Properties

Label 2-690-5.3-c2-0-41
Degree $2$
Conductor $690$
Sign $-0.489 + 0.871i$
Analytic cond. $18.8011$
Root an. cond. $4.33602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − i)2-s + (1.22 + 1.22i)3-s − 2i·4-s + (3.38 − 3.68i)5-s + 2.44·6-s + (−0.957 + 0.957i)7-s + (−2 − 2i)8-s + 2.99i·9-s + (−0.296 − 7.06i)10-s − 8.07·11-s + (2.44 − 2.44i)12-s + (−16.4 − 16.4i)13-s + 1.91i·14-s + (8.65 − 0.363i)15-s − 4·16-s + (10.2 − 10.2i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.408 + 0.408i)3-s − 0.5i·4-s + (0.676 − 0.736i)5-s + 0.408·6-s + (−0.136 + 0.136i)7-s + (−0.250 − 0.250i)8-s + 0.333i·9-s + (−0.0296 − 0.706i)10-s − 0.733·11-s + (0.204 − 0.204i)12-s + (−1.26 − 1.26i)13-s + 0.136i·14-s + (0.576 − 0.0242i)15-s − 0.250·16-s + (0.601 − 0.601i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.489 + 0.871i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (553, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.489 + 0.871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.397218656\)
\(L(\frac12)\) \(\approx\) \(2.397218656\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 + (-1.22 - 1.22i)T \)
5 \( 1 + (-3.38 + 3.68i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (0.957 - 0.957i)T - 49iT^{2} \)
11 \( 1 + 8.07T + 121T^{2} \)
13 \( 1 + (16.4 + 16.4i)T + 169iT^{2} \)
17 \( 1 + (-10.2 + 10.2i)T - 289iT^{2} \)
19 \( 1 + 22.2iT - 361T^{2} \)
29 \( 1 + 9.67iT - 841T^{2} \)
31 \( 1 - 56.2T + 961T^{2} \)
37 \( 1 + (21.1 - 21.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 0.341T + 1.68e3T^{2} \)
43 \( 1 + (-10.7 - 10.7i)T + 1.84e3iT^{2} \)
47 \( 1 + (-11.4 + 11.4i)T - 2.20e3iT^{2} \)
53 \( 1 + (-1.59 - 1.59i)T + 2.80e3iT^{2} \)
59 \( 1 - 10.4iT - 3.48e3T^{2} \)
61 \( 1 - 11.0T + 3.72e3T^{2} \)
67 \( 1 + (-86.5 + 86.5i)T - 4.48e3iT^{2} \)
71 \( 1 + 127.T + 5.04e3T^{2} \)
73 \( 1 + (28.1 + 28.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 72.7iT - 6.24e3T^{2} \)
83 \( 1 + (-61.3 - 61.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 58.5iT - 7.92e3T^{2} \)
97 \( 1 + (101. - 101. i)T - 9.40e3iT^{2} \)
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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

−10.00675995543282043268906800601, −9.407296730627400761800749444894, −8.382475954660613988609746997958, −7.47480975145207662558796185959, −6.07611633896390692004846900020, −5.05476465226679844796264321776, −4.71416139053080135689684348532, −3.02138912571588766988354442554, −2.42284218608717938438622122199, −0.64873124753471459380581958481, 1.90371420649523979087954388380, 2.86803072840071934430592363011, 4.02798464663240180213751047854, 5.28731800976442581675700812079, 6.20186833947778668340472115221, 7.01837145935509544041730111561, 7.67895661972733364487070964619, 8.662889449920146814723304923751, 9.836549596796026411127907256694, 10.27782573631964253094919095681

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