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Properties

Label 2-690-15.2-c1-0-10
Degree $2$
Conductor $690$
Sign $0.300 - 0.953i$
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  + (−0.707 + 0.707i)2-s + (−1.42 − 0.987i)3-s − 1.00i·4-s + (2.16 + 0.556i)5-s + (1.70 − 0.307i)6-s + (3.32 + 3.32i)7-s + (0.707 + 0.707i)8-s + (1.04 + 2.81i)9-s + (−1.92 + 1.13i)10-s − 1.84i·11-s + (−0.987 + 1.42i)12-s + (−1.71 + 1.71i)13-s − 4.69·14-s + (−2.53 − 2.93i)15-s − 1.00·16-s + (−0.948 + 0.948i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.821 − 0.570i)3-s − 0.500i·4-s + (0.968 + 0.248i)5-s + (0.695 − 0.125i)6-s + (1.25 + 1.25i)7-s + (0.250 + 0.250i)8-s + (0.349 + 0.936i)9-s + (−0.608 + 0.359i)10-s − 0.557i·11-s + (−0.285 + 0.410i)12-s + (−0.476 + 0.476i)13-s − 1.25·14-s + (−0.653 − 0.756i)15-s − 0.250·16-s + (−0.229 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.300 - 0.953i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902539 + 0.662267i\)
\(L(\frac12)\) \(\approx\) \(0.902539 + 0.662267i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (1.42 + 0.987i)T \)
5 \( 1 + (-2.16 - 0.556i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-3.32 - 3.32i)T + 7iT^{2} \)
11 \( 1 + 1.84iT - 11T^{2} \)
13 \( 1 + (1.71 - 1.71i)T - 13iT^{2} \)
17 \( 1 + (0.948 - 0.948i)T - 17iT^{2} \)
19 \( 1 - 1.25iT - 19T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + 4.74T + 31T^{2} \)
37 \( 1 + (-6.93 - 6.93i)T + 37iT^{2} \)
41 \( 1 - 0.505iT - 41T^{2} \)
43 \( 1 + (-6.74 + 6.74i)T - 43iT^{2} \)
47 \( 1 + (7.17 - 7.17i)T - 47iT^{2} \)
53 \( 1 + (-7.30 - 7.30i)T + 53iT^{2} \)
59 \( 1 - 7.06T + 59T^{2} \)
61 \( 1 - 3.82T + 61T^{2} \)
67 \( 1 + (2.57 + 2.57i)T + 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (3.30 - 3.30i)T - 73iT^{2} \)
79 \( 1 + 7.20iT - 79T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 + 9.35T + 89T^{2} \)
97 \( 1 + (5.09 + 5.09i)T + 97iT^{2} \)
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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

−10.81858254557739649742126983706, −9.651784463905286097901209427468, −8.874363814733484535582026678439, −7.989811948683662530746728410205, −7.10494331504262838821380936220, −6.00136804628156867001955874277, −5.61936056962513773339179554896, −4.73228170780978298045539719507, −2.36168432351579768146298996403, −1.53407105113997077325173304506, 0.834555361909438653192553224579, 2.10075670557790097969580843656, 3.91248211210090846507225361984, 4.77873492353956018396151409071, 5.54140297464075598894594577038, 6.92402669927089244708805999722, 7.62653936873978093352702208778, 8.863388582038397845523982679805, 9.745075557368419709577080710528, 10.27757435562319679085299195887

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