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Properties

Label 2-690-69.68-c1-0-31
Degree $2$
Conductor $690$
Sign $-0.186 - 0.982i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  − i·2-s + (−0.597 − 1.62i)3-s − 4-s − 5-s + (−1.62 + 0.597i)6-s − 3.80i·7-s + i·8-s + (−2.28 + 1.94i)9-s + i·10-s − 3.48·11-s + (0.597 + 1.62i)12-s + 3.19·13-s − 3.80·14-s + (0.597 + 1.62i)15-s + 16-s − 4.76·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.344 − 0.938i)3-s − 0.5·4-s − 0.447·5-s + (−0.663 + 0.243i)6-s − 1.43i·7-s + 0.353i·8-s + (−0.762 + 0.647i)9-s + 0.316i·10-s − 1.05·11-s + (0.172 + 0.469i)12-s + 0.887·13-s − 1.01·14-s + (0.154 + 0.419i)15-s + 0.250·16-s − 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.186 - 0.982i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.186 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233878 + 0.282401i\)
\(L(\frac12)\) \(\approx\) \(0.233878 + 0.282401i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.597 + 1.62i)T \)
5 \( 1 + T \)
23 \( 1 + (-0.785 + 4.73i)T \)
good7 \( 1 + 3.80iT - 7T^{2} \)
11 \( 1 + 3.48T + 11T^{2} \)
13 \( 1 - 3.19T + 13T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 - 1.88iT - 19T^{2} \)
29 \( 1 - 7.50iT - 29T^{2} \)
31 \( 1 + 3.72T + 31T^{2} \)
37 \( 1 - 4.88iT - 37T^{2} \)
41 \( 1 + 3.99iT - 41T^{2} \)
43 \( 1 + 6.96iT - 43T^{2} \)
47 \( 1 - 10.8iT - 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + 7.35iT - 59T^{2} \)
61 \( 1 - 3.31iT - 61T^{2} \)
67 \( 1 + 3.70iT - 67T^{2} \)
71 \( 1 + 9.92iT - 71T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 2.98iT - 97T^{2} \)
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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

−10.30288096591849863633499865493, −8.835288002994581515578228382992, −8.078812222710466209393649632758, −7.25687915927897975934632828369, −6.44949177220387028487035348140, −5.13707017752271864909179542752, −4.13899347775565793975298448135, −2.97718164800625148147050918512, −1.53067880970679835454150746718, −0.20313434093206486597765129196, 2.62151847142217238909736487336, 3.86958191320295732089533238149, 4.94407703977717673641208496306, 5.65679775348252419553306219394, 6.40818318471034994327256802962, 7.74819220646292668989195653468, 8.662681701699241122438024510257, 9.129322544541121880549129847020, 10.10390197603396066936565955878, 11.22402879375091848076539790079

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