Properties

Label 32-768e16-1.1-c2e16-0-1
Degree $32$
Conductor $1.465\times 10^{46}$
Sign $1$
Analytic cond. $1.35250\times 10^{21}$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·7-s − 160·25-s − 112·31-s − 192·49-s + 160·73-s − 816·79-s + 36·81-s + 192·97-s − 976·103-s − 960·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.26e3·169-s + 173-s + 2.56e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯
L(s)  = 1  − 2.28·7-s − 6.39·25-s − 3.61·31-s − 3.91·49-s + 2.19·73-s − 10.3·79-s + 4/9·81-s + 1.97·97-s − 9.47·103-s − 7.93·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 7.47·169-s + 0.00578·173-s + 14.6·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{128} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{128} \cdot 3^{16}\)
Sign: $1$
Analytic conductor: \(1.35250\times 10^{21}\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{768} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{128} \cdot 3^{16} ,\ ( \ : [1]^{16} ),\ 1 )\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02115548112\)
\(L(\frac12)\) \(\approx\) \(0.02115548112\)
\(L(2)\) 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 \( 1 - 4 p^{2} T^{4} - 512 T^{6} + 470 p^{2} T^{8} - 512 p^{4} T^{10} - 4 p^{10} T^{12} + p^{16} T^{16} \)
good5 \( ( 1 + 16 p T^{2} + 3004 T^{4} + 18672 p T^{6} + 2634054 T^{8} + 18672 p^{5} T^{10} + 3004 p^{8} T^{12} + 16 p^{13} T^{14} + p^{16} T^{16} )^{2} \)
7 \( ( 1 + 4 T + 88 T^{2} + 220 T^{3} + 3646 T^{4} + 220 p^{2} T^{5} + 88 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
11 \( ( 1 + 480 T^{2} + 125212 T^{4} + 22531104 T^{6} + 3078264198 T^{8} + 22531104 p^{4} T^{10} + 125212 p^{8} T^{12} + 480 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
13 \( ( 1 - 632 T^{2} + 214876 T^{4} - 54331336 T^{6} + 10620684806 T^{8} - 54331336 p^{4} T^{10} + 214876 p^{8} T^{12} - 632 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
17 \( ( 1 - 1064 T^{2} + 654556 T^{4} - 282654744 T^{6} + 92440824774 T^{8} - 282654744 p^{4} T^{10} + 654556 p^{8} T^{12} - 1064 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
19 \( ( 1 - 1184 T^{2} + 791836 T^{4} - 351676768 T^{6} + 134635543814 T^{8} - 351676768 p^{4} T^{10} + 791836 p^{8} T^{12} - 1184 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
23 \( ( 1 - 1896 T^{2} + 2093020 T^{4} - 1543350744 T^{6} + 914302437702 T^{8} - 1543350744 p^{4} T^{10} + 2093020 p^{8} T^{12} - 1896 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
29 \( ( 1 + 3024 T^{2} + 4188220 T^{4} + 3460574000 T^{6} + 2571955018182 T^{8} + 3460574000 p^{4} T^{10} + 4188220 p^{8} T^{12} + 3024 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
31 \( ( 1 + 28 T + 3016 T^{2} + 63204 T^{3} + 3999966 T^{4} + 63204 p^{2} T^{5} + 3016 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
37 \( ( 1 - 3640 T^{2} + 6795868 T^{4} - 12043846408 T^{6} + 19395045099910 T^{8} - 12043846408 p^{4} T^{10} + 6795868 p^{8} T^{12} - 3640 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
41 \( ( 1 - 2472 T^{2} + 3015004 T^{4} - 2001918872 T^{6} + 1122734027334 T^{8} - 2001918872 p^{4} T^{10} + 3015004 p^{8} T^{12} - 2472 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
43 \( ( 1 - 8992 T^{2} + 42471580 T^{4} - 131108267744 T^{6} + 285583273079942 T^{8} - 131108267744 p^{4} T^{10} + 42471580 p^{8} T^{12} - 8992 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
47 \( ( 1 - 12296 T^{2} + 72283420 T^{4} - 268133584440 T^{6} + 697058148522822 T^{8} - 268133584440 p^{4} T^{10} + 72283420 p^{8} T^{12} - 12296 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
53 \( ( 1 - 304 T^{2} + 25898428 T^{4} - 6393839056 T^{6} + 288293861465158 T^{8} - 6393839056 p^{4} T^{10} + 25898428 p^{8} T^{12} - 304 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
59 \( ( 1 + 19008 T^{2} + 181109212 T^{4} + 1092163345344 T^{6} + 4537129454477574 T^{8} + 1092163345344 p^{4} T^{10} + 181109212 p^{8} T^{12} + 19008 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
61 \( ( 1 - 19000 T^{2} + 184830940 T^{4} - 1158983771656 T^{6} + 5102256578203654 T^{8} - 1158983771656 p^{4} T^{10} + 184830940 p^{8} T^{12} - 19000 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
67 \( ( 1 - 18112 T^{2} + 165933916 T^{4} - 1017402772288 T^{6} + 4957396059692422 T^{8} - 1017402772288 p^{4} T^{10} + 165933916 p^{8} T^{12} - 18112 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
71 \( ( 1 - 17640 T^{2} + 102948316 T^{4} - 54971476568 T^{6} - 1448379553007034 T^{8} - 54971476568 p^{4} T^{10} + 102948316 p^{8} T^{12} - 17640 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
73 \( ( 1 - 40 T + 15772 T^{2} - 446872 T^{3} + 110402182 T^{4} - 446872 p^{2} T^{5} + 15772 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
79 \( ( 1 + 204 T + 32488 T^{2} + 3143092 T^{3} + 288852126 T^{4} + 3143092 p^{2} T^{5} + 32488 p^{4} T^{6} + 204 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
83 \( ( 1 + 41696 T^{2} + 833769628 T^{4} + 10294236337952 T^{6} + 85511640874023430 T^{8} + 10294236337952 p^{4} T^{10} + 833769628 p^{8} T^{12} + 41696 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
89 \( ( 1 - 17736 T^{2} + 318384412 T^{4} - 3335315895288 T^{6} + 31854516109881798 T^{8} - 3335315895288 p^{4} T^{10} + 318384412 p^{8} T^{12} - 17736 p^{12} T^{14} + p^{16} T^{16} )^{2} \)
97 \( ( 1 - 48 T + 29500 T^{2} - 1080784 T^{3} + 386488710 T^{4} - 1080784 p^{2} T^{5} + 29500 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{4} \)
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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

−2.52833726838414567221157829907, −2.38408906353111682397969333231, −2.34934969621310695104976107935, −2.32550387668540711405197787135, −2.31640346864544360422488048752, −2.00425504135904985675549338902, −1.90455902363982241280563760747, −1.72353326638303979384669443132, −1.69003526420027939328976568608, −1.60715664177229032040707782143, −1.56768069429260979583936373307, −1.52919467581721838306730838760, −1.49383370940795857436527369356, −1.43675760688999368902591272299, −1.40300960928038442737086773858, −1.21843955136389648203169651528, −1.06259400496205542967669613127, −1.05547013862714048927529837345, −0.77606310281353952712315403517, −0.41160100102280899906025713768, −0.30628457991941962999629063557, −0.26488492040350664953393411886, −0.13683541060027609475154013236, −0.12440085083575591172986601742, −0.087397698653973235031812181804, 0.087397698653973235031812181804, 0.12440085083575591172986601742, 0.13683541060027609475154013236, 0.26488492040350664953393411886, 0.30628457991941962999629063557, 0.41160100102280899906025713768, 0.77606310281353952712315403517, 1.05547013862714048927529837345, 1.06259400496205542967669613127, 1.21843955136389648203169651528, 1.40300960928038442737086773858, 1.43675760688999368902591272299, 1.49383370940795857436527369356, 1.52919467581721838306730838760, 1.56768069429260979583936373307, 1.60715664177229032040707782143, 1.69003526420027939328976568608, 1.72353326638303979384669443132, 1.90455902363982241280563760747, 2.00425504135904985675549338902, 2.31640346864544360422488048752, 2.32550387668540711405197787135, 2.34934969621310695104976107935, 2.38408906353111682397969333231, 2.52833726838414567221157829907

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

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