Properties

Label 32-2e160-1.1-c1e16-0-0
Degree $32$
Conductor $1.462\times 10^{48}$
Sign $1$
Analytic cond. $3.99239\times 10^{14}$
Root an. cond. $2.85948$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·5-s + 8·9-s + 24·13-s + 48·25-s − 8·29-s + 8·37-s + 16·41-s − 64·45-s + 40·53-s + 8·61-s − 192·65-s − 32·73-s + 32·81-s − 32·89-s − 16·97-s − 24·101-s + 56·109-s + 192·117-s − 8·121-s − 200·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 3.57·5-s + 8/3·9-s + 6.65·13-s + 48/5·25-s − 1.48·29-s + 1.31·37-s + 2.49·41-s − 9.54·45-s + 5.49·53-s + 1.02·61-s − 23.8·65-s − 3.74·73-s + 32/9·81-s − 3.39·89-s − 1.62·97-s − 2.38·101-s + 5.36·109-s + 17.7·117-s − 0.727·121-s − 17.8·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{160}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{160}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{160}\)
Sign: $1$
Analytic conductor: \(3.99239\times 10^{14}\)
Root analytic conductor: \(2.85948\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{160} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.822988341\)
\(L(\frac12)\) \(\approx\) \(2.822988341\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 8 T^{2} + 32 T^{4} - 152 T^{6} + 604 T^{8} - 1672 T^{10} + 5600 T^{12} - 16984 T^{14} + 42694 T^{16} - 16984 p^{2} T^{18} + 5600 p^{4} T^{20} - 1672 p^{6} T^{22} + 604 p^{8} T^{24} - 152 p^{10} T^{26} + 32 p^{12} T^{28} - 8 p^{14} T^{30} + p^{16} T^{32} \)
5 \( ( 1 + 4 T - 28 T^{3} - 48 T^{4} + 12 p T^{5} + 256 T^{6} - 4 p^{2} T^{7} - 1266 T^{8} - 4 p^{3} T^{9} + 256 p^{2} T^{10} + 12 p^{4} T^{11} - 48 p^{4} T^{12} - 28 p^{5} T^{13} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
7 \( 1 - 24 T^{4} - 2564 T^{8} + 76248 T^{12} + 4414470 T^{16} + 76248 p^{4} T^{20} - 2564 p^{8} T^{24} - 24 p^{12} T^{28} + p^{16} T^{32} \)
11 \( 1 + 8 T^{2} + 32 T^{4} + 3032 T^{6} + 16732 T^{8} - 287416 T^{10} + 14560 p^{2} T^{12} - 280824 p T^{14} - 822624186 T^{16} - 280824 p^{3} T^{18} + 14560 p^{6} T^{20} - 287416 p^{6} T^{22} + 16732 p^{8} T^{24} + 3032 p^{10} T^{26} + 32 p^{12} T^{28} + 8 p^{14} T^{30} + p^{16} T^{32} \)
13 \( ( 1 - 12 T + 64 T^{2} - 12 p T^{3} - 112 T^{4} + 2364 T^{5} - 8832 T^{6} + 12972 T^{7} - 5554 T^{8} + 12972 p T^{9} - 8832 p^{2} T^{10} + 2364 p^{3} T^{11} - 112 p^{4} T^{12} - 12 p^{6} T^{13} + 64 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
17 \( ( 1 - 64 T^{2} + 2412 T^{4} - 61760 T^{6} + 1207526 T^{8} - 61760 p^{2} T^{10} + 2412 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
19 \( 1 + 24 T^{2} + 288 T^{4} + 20040 T^{6} + 447196 T^{8} + 6866712 T^{10} + 236809440 T^{12} + 4556359944 T^{14} + 61111615686 T^{16} + 4556359944 p^{2} T^{18} + 236809440 p^{4} T^{20} + 6866712 p^{6} T^{22} + 447196 p^{8} T^{24} + 20040 p^{10} T^{26} + 288 p^{12} T^{28} + 24 p^{14} T^{30} + p^{16} T^{32} \)
23 \( 1 - 1368 T^{4} + 659068 T^{8} - 135138024 T^{12} + 27188126214 T^{16} - 135138024 p^{4} T^{20} + 659068 p^{8} T^{24} - 1368 p^{12} T^{28} + p^{16} T^{32} \)
29 \( ( 1 + 4 T + 64 T^{2} - 92 T^{3} + 720 T^{4} - 15908 T^{5} + 22592 T^{6} - 377540 T^{7} + 1763918 T^{8} - 377540 p T^{9} + 22592 p^{2} T^{10} - 15908 p^{3} T^{11} + 720 p^{4} T^{12} - 92 p^{5} T^{13} + 64 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
31 \( ( 1 + 24 T^{2} + 924 T^{4} + 32424 T^{6} - 181690 T^{8} + 32424 p^{2} T^{10} + 924 p^{4} T^{12} + 24 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 4 T + 48 T^{2} - 420 T^{3} + 2128 T^{4} - 12380 T^{5} + 118608 T^{6} - 567612 T^{7} + 4122126 T^{8} - 567612 p T^{9} + 118608 p^{2} T^{10} - 12380 p^{3} T^{11} + 2128 p^{4} T^{12} - 420 p^{5} T^{13} + 48 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 8 T + 32 T^{2} - 184 T^{3} + 1628 T^{4} - 10216 T^{5} + 46560 T^{6} - 75224 T^{7} - 1980282 T^{8} - 75224 p T^{9} + 46560 p^{2} T^{10} - 10216 p^{3} T^{11} + 1628 p^{4} T^{12} - 184 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
43 \( 1 + 280 T^{2} + 39200 T^{4} + 3734920 T^{6} + 275251804 T^{8} + 16805127640 T^{10} + 890378725600 T^{12} + 42533679223560 T^{14} + 1886841963508806 T^{16} + 42533679223560 p^{2} T^{18} + 890378725600 p^{4} T^{20} + 16805127640 p^{6} T^{22} + 275251804 p^{8} T^{24} + 3734920 p^{10} T^{26} + 39200 p^{12} T^{28} + 280 p^{14} T^{30} + p^{16} T^{32} \)
47 \( ( 1 - 120 T^{2} + 7644 T^{4} - 420168 T^{6} + 22246982 T^{8} - 420168 p^{2} T^{10} + 7644 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
53 \( ( 1 - 20 T + 160 T^{2} + 204 T^{3} - 15280 T^{4} + 130420 T^{5} - 196320 T^{6} - 5901292 T^{7} + 68076814 T^{8} - 5901292 p T^{9} - 196320 p^{2} T^{10} + 130420 p^{3} T^{11} - 15280 p^{4} T^{12} + 204 p^{5} T^{13} + 160 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( 1 - 216 T^{2} + 23328 T^{4} - 1766664 T^{6} + 107825500 T^{8} - 92970504 p T^{10} + 230013683424 T^{12} - 6568095595464 T^{14} + 171368345746758 T^{16} - 6568095595464 p^{2} T^{18} + 230013683424 p^{4} T^{20} - 92970504 p^{7} T^{22} + 107825500 p^{8} T^{24} - 1766664 p^{10} T^{26} + 23328 p^{12} T^{28} - 216 p^{14} T^{30} + p^{16} T^{32} \)
61 \( ( 1 - 4 T - 128 T^{2} + 220 T^{3} + 9424 T^{4} - 14780 T^{5} - 123328 T^{6} - 183004 T^{7} - 2998578 T^{8} - 183004 p T^{9} - 123328 p^{2} T^{10} - 14780 p^{3} T^{11} + 9424 p^{4} T^{12} + 220 p^{5} T^{13} - 128 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
67 \( 1 - 24 T^{2} + 288 T^{4} + 196536 T^{6} - 46292900 T^{8} + 466437096 T^{10} + 21451064544 T^{12} - 4119669415944 T^{14} + 1054597662328518 T^{16} - 4119669415944 p^{2} T^{18} + 21451064544 p^{4} T^{20} + 466437096 p^{6} T^{22} - 46292900 p^{8} T^{24} + 196536 p^{10} T^{26} + 288 p^{12} T^{28} - 24 p^{14} T^{30} + p^{16} T^{32} \)
71 \( 1 + 6952 T^{4} + 30802044 T^{8} - 214228188520 T^{12} - 1434458600506618 T^{16} - 214228188520 p^{4} T^{20} + 30802044 p^{8} T^{24} + 6952 p^{12} T^{28} + p^{16} T^{32} \)
73 \( ( 1 + 16 T + 128 T^{2} + 1264 T^{3} + 10972 T^{4} + 51664 T^{5} + 221056 T^{6} - 1001424 T^{7} - 36752442 T^{8} - 1001424 p T^{9} + 221056 p^{2} T^{10} + 51664 p^{3} T^{11} + 10972 p^{4} T^{12} + 1264 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
79 \( ( 1 - 504 T^{2} + 119004 T^{4} - 17120712 T^{6} + 1638907718 T^{8} - 17120712 p^{2} T^{10} + 119004 p^{4} T^{12} - 504 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( 1 + 168 T^{2} + 14112 T^{4} + 1906488 T^{6} + 181015900 T^{8} + 9600901608 T^{10} + 875803336416 T^{12} + 61358369464248 T^{14} + 2408049709874118 T^{16} + 61358369464248 p^{2} T^{18} + 875803336416 p^{4} T^{20} + 9600901608 p^{6} T^{22} + 181015900 p^{8} T^{24} + 1906488 p^{10} T^{26} + 14112 p^{12} T^{28} + 168 p^{14} T^{30} + p^{16} T^{32} \)
89 \( ( 1 + 16 T + 128 T^{2} + 1616 T^{3} + 28316 T^{4} + 281264 T^{5} + 2181504 T^{6} + 28123120 T^{7} + 359358150 T^{8} + 28123120 p T^{9} + 2181504 p^{2} T^{10} + 281264 p^{3} T^{11} + 28316 p^{4} T^{12} + 1616 p^{5} T^{13} + 128 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
97 \( ( 1 + 4 T + 240 T^{2} + 860 T^{3} + 28118 T^{4} + 860 p T^{5} + 240 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−2.71769956209730124581231238909, −2.64086607443740000617355216685, −2.44212908017127468188820011161, −2.42466371049021364821495857790, −2.31593372816468038416956041172, −2.06471952638371828822634769418, −1.94693064198729849934727754520, −1.91586969638388108079425036996, −1.89328543413997739222236572975, −1.80773245162903649241474507679, −1.68747077609608571577665776323, −1.61667918743200463466502080484, −1.60795260100134648820432865054, −1.46548427113500013714643873729, −1.25154402252117039907274634508, −1.07584810737702601775278411457, −1.06350014733575390134787636616, −1.05862393587173426842759717704, −0.987721438073612037700659298721, −0.915678475219895319754870054895, −0.824889174152015938381849805190, −0.70388115667100656805078070210, −0.59911325175257719618924307879, −0.40931716648352209947093104492, −0.05813472493575223112654162793, 0.05813472493575223112654162793, 0.40931716648352209947093104492, 0.59911325175257719618924307879, 0.70388115667100656805078070210, 0.824889174152015938381849805190, 0.915678475219895319754870054895, 0.987721438073612037700659298721, 1.05862393587173426842759717704, 1.06350014733575390134787636616, 1.07584810737702601775278411457, 1.25154402252117039907274634508, 1.46548427113500013714643873729, 1.60795260100134648820432865054, 1.61667918743200463466502080484, 1.68747077609608571577665776323, 1.80773245162903649241474507679, 1.89328543413997739222236572975, 1.91586969638388108079425036996, 1.94693064198729849934727754520, 2.06471952638371828822634769418, 2.31593372816468038416956041172, 2.42466371049021364821495857790, 2.44212908017127468188820011161, 2.64086607443740000617355216685, 2.71769956209730124581231238909

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.