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Properties

Label 32-420e16-1.1-c1e16-0-3
Degree $32$
Conductor $9.375\times 10^{41}$
Sign $1$
Analytic cond. $2.56107\times 10^{8}$
Root an. cond. $1.83131$
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  + 16·11-s + 8·23-s + 8·25-s + 16·37-s + 48·43-s − 40·53-s − 48·67-s − 32·71-s − 4·81-s − 56·107-s + 72·113-s + 16·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  + 4.82·11-s + 1.66·23-s + 8/5·25-s + 2.63·37-s + 7.31·43-s − 5.49·53-s − 5.86·67-s − 3.79·71-s − 4/9·81-s − 5.41·107-s + 6.77·113-s + 1.45·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(2.56107\times 10^{8}\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{16} \cdot 5^{16} \cdot 7^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(11.46723809\)
\(L(\frac12)\) \(\approx\) \(11.46723809\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( ( 1 + T^{4} )^{4} \)
5 \( 1 - 8 T^{2} - 4 T^{4} + 8 p^{2} T^{6} - 986 T^{8} + 8 p^{4} T^{10} - 4 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \)
7 \( 1 + 32 T^{3} + 36 T^{4} + 96 T^{5} + 512 T^{6} + 1216 T^{7} + 3334 T^{8} + 1216 p T^{9} + 512 p^{2} T^{10} + 96 p^{3} T^{11} + 36 p^{4} T^{12} + 32 p^{5} T^{13} + p^{8} T^{16} \)
good11 \( ( 1 - 4 T + 36 T^{2} - 116 T^{3} + 546 T^{4} - 116 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
13 \( 1 - 536 T^{4} + 97308 T^{8} - 1310248 T^{12} - 9274858 p^{2} T^{16} - 1310248 p^{4} T^{20} + 97308 p^{8} T^{24} - 536 p^{12} T^{28} + p^{16} T^{32} \)
17 \( 1 - 904 T^{4} + 210332 T^{8} + 67692360 T^{12} - 44568692154 T^{16} + 67692360 p^{4} T^{20} + 210332 p^{8} T^{24} - 904 p^{12} T^{28} + p^{16} T^{32} \)
19 \( ( 1 + 120 T^{2} + 6692 T^{4} + 228264 T^{6} + 5227942 T^{8} + 228264 p^{2} T^{10} + 6692 p^{4} T^{12} + 120 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
23 \( ( 1 - 4 T + 8 T^{2} + 4 T^{3} - 1548 T^{4} + 172 p T^{5} - 3432 T^{6} - 53556 T^{7} + 1130086 T^{8} - 53556 p T^{9} - 3432 p^{2} T^{10} + 172 p^{4} T^{11} - 1548 p^{4} T^{12} + 4 p^{5} T^{13} + 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
29 \( ( 1 - 136 T^{2} + 8924 T^{4} - 381688 T^{6} + 12377958 T^{8} - 381688 p^{2} T^{10} + 8924 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
31 \( ( 1 - 136 T^{2} + 8612 T^{4} - 354840 T^{6} + 11712390 T^{8} - 354840 p^{2} T^{10} + 8612 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
37 \( ( 1 - 8 T + 32 T^{2} - 56 T^{3} - 2980 T^{4} + 15992 T^{5} - 31008 T^{6} - 263416 T^{7} + 5332518 T^{8} - 263416 p T^{9} - 31008 p^{2} T^{10} + 15992 p^{3} T^{11} - 2980 p^{4} T^{12} - 56 p^{5} T^{13} + 32 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
41 \( ( 1 - 200 T^{2} + 20668 T^{4} - 1400376 T^{6} + 67532294 T^{8} - 1400376 p^{2} T^{10} + 20668 p^{4} T^{12} - 200 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
43 \( ( 1 - 24 T + 288 T^{2} - 2504 T^{3} + 19620 T^{4} - 145992 T^{5} + 988256 T^{6} - 5831128 T^{7} + 34926502 T^{8} - 5831128 p T^{9} + 988256 p^{2} T^{10} - 145992 p^{3} T^{11} + 19620 p^{4} T^{12} - 2504 p^{5} T^{13} + 288 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
47 \( 1 + 2312 T^{4} - 7152868 T^{8} - 8327985864 T^{12} + 35781048692550 T^{16} - 8327985864 p^{4} T^{20} - 7152868 p^{8} T^{24} + 2312 p^{12} T^{28} + p^{16} T^{32} \)
53 \( ( 1 + 20 T + 200 T^{2} + 1284 T^{3} + 2996 T^{4} - 21028 T^{5} - 195432 T^{6} - 604308 T^{7} - 304634 T^{8} - 604308 p T^{9} - 195432 p^{2} T^{10} - 21028 p^{3} T^{11} + 2996 p^{4} T^{12} + 1284 p^{5} T^{13} + 200 p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
59 \( ( 1 + 280 T^{2} + 33212 T^{4} + 2343784 T^{6} + 136126182 T^{8} + 2343784 p^{2} T^{10} + 33212 p^{4} T^{12} + 280 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
61 \( ( 1 - 312 T^{2} + 47324 T^{4} - 4644168 T^{6} + 328520806 T^{8} - 4644168 p^{2} T^{10} + 47324 p^{4} T^{12} - 312 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
67 \( ( 1 + 24 T + 288 T^{2} + 2984 T^{3} + 22788 T^{4} + 56616 T^{5} - 752032 T^{6} - 14609960 T^{7} - 156156506 T^{8} - 14609960 p T^{9} - 752032 p^{2} T^{10} + 56616 p^{3} T^{11} + 22788 p^{4} T^{12} + 2984 p^{5} T^{13} + 288 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} )^{2} \)
71 \( ( 1 + 8 T + 140 T^{2} + 1024 T^{3} + 14962 T^{4} + 1024 p T^{5} + 140 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{4} \)
73 \( 1 - 9816 T^{4} + 48230108 T^{8} - 385344995944 T^{12} + 2907384584774982 T^{16} - 385344995944 p^{4} T^{20} + 48230108 p^{8} T^{24} - 9816 p^{12} T^{28} + p^{16} T^{32} \)
79 \( ( 1 - 248 T^{2} + 39836 T^{4} - 4744648 T^{6} + 419330758 T^{8} - 4744648 p^{2} T^{10} + 39836 p^{4} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
83 \( 1 + 11592 T^{4} + 91463132 T^{8} + 885997919352 T^{12} + 6666213559678726 T^{16} + 885997919352 p^{4} T^{20} + 91463132 p^{8} T^{24} + 11592 p^{12} T^{28} + p^{16} T^{32} \)
89 \( ( 1 + 328 T^{2} + 42716 T^{4} + 2964280 T^{6} + 188103750 T^{8} + 2964280 p^{2} T^{10} + 42716 p^{4} T^{12} + 328 p^{6} T^{14} + p^{8} T^{16} )^{2} \)
97 \( 1 - 1624 T^{4} + 150045276 T^{8} - 609733450088 T^{12} + 17981575892543942 T^{16} - 609733450088 p^{4} T^{20} + 150045276 p^{8} T^{24} - 1624 p^{12} T^{28} + p^{16} T^{32} \)
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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

−3.09426375726172007609234322185, −3.07913783461317167010466037619, −3.07604346919367524039017865194, −2.95382653833362114420067477143, −2.67787103379285600606130733487, −2.58640533181702755155435827981, −2.55638495979039919863317621472, −2.53482182681494907340450791484, −2.48062630238606465766564799399, −2.43161056196125646428979578890, −2.20355141403170412749390489536, −1.89717824871004904157444755026, −1.86645626513701778656946841679, −1.84185057190328491221067946299, −1.74019794928679268226954019277, −1.52000898179056393986423328358, −1.48586975329676860564442674987, −1.32358217688091225818719237401, −1.21287348953837831029189543098, −1.16901268533935566988902774569, −1.06681607952319683181669337991, −0.982288810155723063243244256966, −0.935112517148775688389511860997, −0.42541413721948743761755276086, −0.29037817382629598321791135629, 0.29037817382629598321791135629, 0.42541413721948743761755276086, 0.935112517148775688389511860997, 0.982288810155723063243244256966, 1.06681607952319683181669337991, 1.16901268533935566988902774569, 1.21287348953837831029189543098, 1.32358217688091225818719237401, 1.48586975329676860564442674987, 1.52000898179056393986423328358, 1.74019794928679268226954019277, 1.84185057190328491221067946299, 1.86645626513701778656946841679, 1.89717824871004904157444755026, 2.20355141403170412749390489536, 2.43161056196125646428979578890, 2.48062630238606465766564799399, 2.53482182681494907340450791484, 2.55638495979039919863317621472, 2.58640533181702755155435827981, 2.67787103379285600606130733487, 2.95382653833362114420067477143, 3.07604346919367524039017865194, 3.07913783461317167010466037619, 3.09426375726172007609234322185

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

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