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Properties

Label 2-690-345.344-c1-0-3
Degree $2$
Conductor $690$
Sign $0.0506 - 0.998i$
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 + (0.571 − 1.63i)3-s + 4-s + (−2.12 + 0.689i)5-s + (−0.571 + 1.63i)6-s + 1.28·7-s − 8-s + (−2.34 − 1.86i)9-s + (2.12 − 0.689i)10-s − 4.52·11-s + (0.571 − 1.63i)12-s + 3.04i·13-s − 1.28·14-s + (−0.0895 + 3.87i)15-s + 16-s + 5.73i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.330 − 0.943i)3-s + 0.5·4-s + (−0.951 + 0.308i)5-s + (−0.233 + 0.667i)6-s + 0.487·7-s − 0.353·8-s + (−0.781 − 0.623i)9-s + (0.672 − 0.217i)10-s − 1.36·11-s + (0.165 − 0.471i)12-s + 0.843i·13-s − 0.344·14-s + (−0.0231 + 0.999i)15-s + 0.250·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0506 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0506 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0506 - 0.998i$
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: \((2,\ 690,\ (\ :1/2),\ 0.0506 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348075 + 0.330863i\)
\(L(\frac12)\) \(\approx\) \(0.348075 + 0.330863i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.571 + 1.63i)T \)
5 \( 1 + (2.12 - 0.689i)T \)
23 \( 1 + (-4.79 - 0.132i)T \)
good7 \( 1 - 1.28T + 7T^{2} \)
11 \( 1 + 4.52T + 11T^{2} \)
13 \( 1 - 3.04iT - 13T^{2} \)
17 \( 1 - 5.73iT - 17T^{2} \)
19 \( 1 - 3.52iT - 19T^{2} \)
29 \( 1 + 3.03iT - 29T^{2} \)
31 \( 1 + 7.30T + 31T^{2} \)
37 \( 1 + 8.69T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 8.85T + 43T^{2} \)
47 \( 1 - 3.48T + 47T^{2} \)
53 \( 1 - 4.54iT - 53T^{2} \)
59 \( 1 + 12.7iT - 59T^{2} \)
61 \( 1 - 3.04iT - 61T^{2} \)
67 \( 1 - 3.98T + 67T^{2} \)
71 \( 1 - 5.30iT - 71T^{2} \)
73 \( 1 - 9.43iT - 73T^{2} \)
79 \( 1 - 16.0iT - 79T^{2} \)
83 \( 1 - 8.09iT - 83T^{2} \)
89 \( 1 + 6.63T + 89T^{2} \)
97 \( 1 - 0.0268T + 97T^{2} \)
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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

−10.99808260854753880928693661426, −9.746755805225708995443279372635, −8.528231696173604253309821922369, −8.129043745815754558143115902502, −7.41273073552630626824391045897, −6.62820659218562935623399787517, −5.47875595735205489073948127123, −3.94304207622509414601332600770, −2.74650801367295134553962900075, −1.54551995411942217029505031433, 0.30212827312400123083490664207, 2.57928140880106070854107843653, 3.48779544595335922147906792314, 4.96100545967908526853988002053, 5.30546061835617715775645160349, 7.33979198821033178494464760563, 7.65026185288209198838962687309, 8.813998872511541306137108799178, 9.082234841512609766705122990207, 10.49848858555403045573720048515

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