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Properties

Label 2-690-345.344-c1-0-13
Degree $2$
Conductor $690$
Sign $0.427 - 0.904i$
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  − 2-s + (1.68 + 0.401i)3-s + 4-s + (1.83 + 1.28i)5-s + (−1.68 − 0.401i)6-s − 1.73·7-s − 8-s + (2.67 + 1.35i)9-s + (−1.83 − 1.28i)10-s − 1.14·11-s + (1.68 + 0.401i)12-s + 5.69i·13-s + 1.73·14-s + (2.56 + 2.89i)15-s + 16-s − 5.08i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.972 + 0.231i)3-s + 0.5·4-s + (0.818 + 0.574i)5-s + (−0.687 − 0.163i)6-s − 0.656·7-s − 0.353·8-s + (0.892 + 0.450i)9-s + (−0.578 − 0.406i)10-s − 0.344·11-s + (0.486 + 0.115i)12-s + 1.57i·13-s + 0.463·14-s + (0.663 + 0.748i)15-s + 0.250·16-s − 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.427 - 0.904i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.427 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36009 + 0.861805i\)
\(L(\frac12)\) \(\approx\) \(1.36009 + 0.861805i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (-1.68 - 0.401i)T \)
5 \( 1 + (-1.83 - 1.28i)T \)
23 \( 1 + (-1.88 - 4.40i)T \)
good7 \( 1 + 1.73T + 7T^{2} \)
11 \( 1 + 1.14T + 11T^{2} \)
13 \( 1 - 5.69iT - 13T^{2} \)
17 \( 1 + 5.08iT - 17T^{2} \)
19 \( 1 - 4.40iT - 19T^{2} \)
29 \( 1 + 4.89iT - 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 + 5.25iT - 41T^{2} \)
43 \( 1 - 3.19T + 43T^{2} \)
47 \( 1 - 2.36T + 47T^{2} \)
53 \( 1 + 7.04iT - 53T^{2} \)
59 \( 1 + 1.33iT - 59T^{2} \)
61 \( 1 + 7.49iT - 61T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 4.45iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 4.09iT - 83T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 - 4.22T + 97T^{2} \)
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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

−10.17207024842947005474102593996, −9.677753656100703240558640855623, −9.171175376431878386830621590940, −8.165749010139163774942509183149, −7.09072644864290907557828162539, −6.60507937576798001104632157998, −5.26072301401019019363994349655, −3.77802142667879985204145489198, −2.72873074568365743017383807779, −1.79816114014669676883696583063, 1.00136186500540443992466009600, 2.42877509739018271576599974957, 3.27271948501757640635165027496, 4.86733971686142205728778355391, 6.08529954432321168942189746489, 6.89417151499730212498725202310, 8.046337598940503415443666831452, 8.607704350203198218048241376262, 9.311128914809196707552566949910, 10.31662617694814509279826307735

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