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Properties

Label 2-696-232.99-c1-0-41
Degree $2$
Conductor $696$
Sign $0.828 - 0.560i$
Analytic cond. $5.55758$
Root an. cond. $2.35745$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(696\)    =    \(2^{3} \cdot 3 \cdot 29\)
Sign: $0.828 - 0.560i$
Analytic conductor: \(5.55758\)
Root analytic conductor: \(2.35745\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{696} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 696,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.25518 + 0.998296i\)
\(L(\frac12)\) \(\approx\) \(3.25518 + 0.998296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

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 $Z$-function along the critical line