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Properties

Label 2-696-232.99-c1-0-41
Degree 22
Conductor 696696
Sign 0.828−0.560i0.828 - 0.560i
Analytic cond. 5.557585.55758
Root an. cond. 2.357452.35745
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + (0.707 + 0.707i)3-s + 2.00·4-s + 5-s + (1.00 + 1.00i)6-s + 2.41i·7-s + 2.82·8-s + 1.00i·9-s + 1.41·10-s + (−1.82 − 1.82i)11-s + (1.41 + 1.41i)12-s + 2.82·13-s + 3.41i·14-s + (0.707 + 0.707i)15-s + 4.00·16-s + (−5.53 − 5.53i)17-s + ⋯
L(s)  = 1  + 1.00·2-s + (0.408 + 0.408i)3-s + 1.00·4-s + 0.447·5-s + (0.408 + 0.408i)6-s + 0.912i·7-s + 1.00·8-s + 0.333i·9-s + 0.447·10-s + (−0.551 − 0.551i)11-s + (0.408 + 0.408i)12-s + 0.784·13-s + 0.912i·14-s + (0.182 + 0.182i)15-s + 1.00·16-s + (−1.34 − 1.34i)17-s + ⋯

Functional equation

Λ(s)=(696s/2 ΓC(s) L(s)=((0.828−0.560i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(696s/2 ΓC(s+1/2) L(s)=((0.828−0.560i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 696696    =    23⋅3⋅292^{3} \cdot 3 \cdot 29
Sign: 0.828−0.560i0.828 - 0.560i
Analytic conductor: 5.557585.55758
Root analytic conductor: 2.357452.35745
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ696(331,⋅)\chi_{696} (331, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 696, ( :1/2), 0.828−0.560i)(2,\ 696,\ (\ :1/2),\ 0.828 - 0.560i)

Particular Values

L(1)L(1) ≈\approx 3.25518+0.998296i3.25518 + 0.998296i
L(12)L(\frac12) ≈\approx 3.25518+0.998296i3.25518 + 0.998296i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1−1.41T 1 - 1.41T
3 1+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
29 1+(2−5i)T 1 + (2 - 5i)T
good5 1−T+5T2 1 - T + 5T^{2}
7 1−2.41iT−7T2 1 - 2.41iT - 7T^{2}
11 1+(1.82+1.82i)T+11iT2 1 + (1.82 + 1.82i)T + 11iT^{2}
13 1−2.82T+13T2 1 - 2.82T + 13T^{2}
17 1+(5.53+5.53i)T+17iT2 1 + (5.53 + 5.53i)T + 17iT^{2}
19 1+(−1.70−1.70i)T+19iT2 1 + (-1.70 - 1.70i)T + 19iT^{2}
23 1−4.58iT−23T2 1 - 4.58iT - 23T^{2}
31 1+(−3.58+3.58i)T−31iT2 1 + (-3.58 + 3.58i)T - 31iT^{2}
37 1+(3.29+3.29i)T+37iT2 1 + (3.29 + 3.29i)T + 37iT^{2}
41 1+(−6.12+6.12i)T−41iT2 1 + (-6.12 + 6.12i)T - 41iT^{2}
43 1+(1.94+1.94i)T+43iT2 1 + (1.94 + 1.94i)T + 43iT^{2}
47 1+(2.29+2.29i)T+47iT2 1 + (2.29 + 2.29i)T + 47iT^{2}
53 1+8.48iT−53T2 1 + 8.48iT - 53T^{2}
59 1+8.89T+59T2 1 + 8.89T + 59T^{2}
61 1+(2.58−2.58i)T−61iT2 1 + (2.58 - 2.58i)T - 61iT^{2}
67 1−5.75iT−67T2 1 - 5.75iT - 67T^{2}
71 1−12.8T+71T2 1 - 12.8T + 71T^{2}
73 1+(1.82−1.82i)T−73iT2 1 + (1.82 - 1.82i)T - 73iT^{2}
79 1+(0.828−0.828i)T−79iT2 1 + (0.828 - 0.828i)T - 79iT^{2}
83 1−2.34T+83T2 1 - 2.34T + 83T^{2}
89 1+(8.24+8.24i)T+89iT2 1 + (8.24 + 8.24i)T + 89iT^{2}
97 1+(2.24−2.24i)T−97iT2 1 + (2.24 - 2.24i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.78244991474135468363539010291, −9.649807690054865515726945748832, −8.897027305421876454582966870592, −7.893520491839913751749354681120, −6.82586769155836746850835733221, −5.70138218571883909827450228334, −5.26403638719014980775482782678, −3.98021033900875579890087531424, −2.94637533369019403479280709082, −2.02395862555363730847030694482, 1.56452491471711705802027108721, 2.66437081472724404089743735736, 3.94376458876010435572631386661, 4.67596716088768822728506767880, 6.09266880825976277238189038985, 6.60418209703328800452815853141, 7.63054040680991106252891650520, 8.383836637147653057980980354212, 9.696333877930266003926244942552, 10.62119802492323490696048588012

Graph of the ZZ-function along the critical line