Properties

Label 2-690-15.8-c1-0-39
Degree $2$
Conductor $690$
Sign $-0.899 + 0.436i$
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 + (0.451 − 1.67i)3-s + 1.00i·4-s + (0.664 − 2.13i)5-s + (−1.50 + 0.862i)6-s + (2.34 − 2.34i)7-s + (0.707 − 0.707i)8-s + (−2.59 − 1.51i)9-s + (−1.97 + 1.03i)10-s − 4.92i·11-s + (1.67 + 0.451i)12-s + (2.78 + 2.78i)13-s − 3.31·14-s + (−3.26 − 2.07i)15-s − 1.00·16-s + (2.05 + 2.05i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (0.260 − 0.965i)3-s + 0.500i·4-s + (0.297 − 0.954i)5-s + (−0.613 + 0.352i)6-s + (0.884 − 0.884i)7-s + (0.250 − 0.250i)8-s + (−0.863 − 0.503i)9-s + (−0.625 + 0.328i)10-s − 1.48i·11-s + (0.482 + 0.130i)12-s + (0.772 + 0.772i)13-s − 0.884·14-s + (−0.844 − 0.536i)15-s − 0.250·16-s + (0.497 + 0.497i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.899 + 0.436i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.899 + 0.436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324606 - 1.41340i\)
\(L(\frac12)\) \(\approx\) \(0.324606 - 1.41340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.451 + 1.67i)T \)
5 \( 1 + (-0.664 + 2.13i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + (-2.34 + 2.34i)T - 7iT^{2} \)
11 \( 1 + 4.92iT - 11T^{2} \)
13 \( 1 + (-2.78 - 2.78i)T + 13iT^{2} \)
17 \( 1 + (-2.05 - 2.05i)T + 17iT^{2} \)
19 \( 1 - 3.73iT - 19T^{2} \)
29 \( 1 - 9.53T + 29T^{2} \)
31 \( 1 + 2.15T + 31T^{2} \)
37 \( 1 + (1.67 - 1.67i)T - 37iT^{2} \)
41 \( 1 - 5.18iT - 41T^{2} \)
43 \( 1 + (-2.73 - 2.73i)T + 43iT^{2} \)
47 \( 1 + (8.67 + 8.67i)T + 47iT^{2} \)
53 \( 1 + (8.92 - 8.92i)T - 53iT^{2} \)
59 \( 1 - 4.44T + 59T^{2} \)
61 \( 1 + 2.83T + 61T^{2} \)
67 \( 1 + (6.60 - 6.60i)T - 67iT^{2} \)
71 \( 1 + 1.39iT - 71T^{2} \)
73 \( 1 + (-7.65 - 7.65i)T + 73iT^{2} \)
79 \( 1 - 1.62iT - 79T^{2} \)
83 \( 1 + (0.357 - 0.357i)T - 83iT^{2} \)
89 \( 1 + 7.92T + 89T^{2} \)
97 \( 1 + (-5.52 + 5.52i)T - 97iT^{2} \)
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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

−10.13868637416428281347516770803, −8.958652795077989869359659714219, −8.285732710187307227048486038813, −7.976729821904154849914447628454, −6.62797736551164182443517470247, −5.74358291928791700659899276474, −4.36880882266678424918756891755, −3.25436254215923092543334961762, −1.59762016135038809888338679595, −0.988328550153633662540079317673, 2.07062208244596513657105089566, 3.13866975288570138025366259931, 4.69615735169674590131131066419, 5.36133123339724617234203576697, 6.42495817791738172583967909926, 7.51330770719259594717016674880, 8.293688952178172885936244625801, 9.201880552833547562661338637551, 9.885169137269582791053896343306, 10.65197217286428031908403775245

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