Properties

Label 2-690-23.16-c1-0-0
Degree $2$
Conductor $690$
Sign $-0.998 + 0.0610i$
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  + (−0.654 + 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (0.170 − 0.0501i)7-s + (0.841 + 0.540i)8-s + (0.415 − 0.909i)9-s + (−0.959 − 0.281i)10-s + (−0.0281 − 0.0324i)11-s + (0.654 + 0.755i)12-s + (−4.70 − 1.38i)13-s + (−0.0739 + 0.161i)14-s + (−0.841 − 0.540i)15-s + (−0.959 + 0.281i)16-s + (−0.635 + 4.41i)17-s + ⋯
L(s)  = 1  + (−0.463 + 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (0.0580 − 0.404i)6-s + (0.0645 − 0.0189i)7-s + (0.297 + 0.191i)8-s + (0.138 − 0.303i)9-s + (−0.303 − 0.0890i)10-s + (−0.00848 − 0.00979i)11-s + (0.189 + 0.218i)12-s + (−1.30 − 0.383i)13-s + (−0.0197 + 0.0432i)14-s + (−0.217 − 0.139i)15-s + (−0.239 + 0.0704i)16-s + (−0.154 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.998 + 0.0610i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.998 + 0.0610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0140437 - 0.459297i\)
\(L(\frac12)\) \(\approx\) \(0.0140437 - 0.459297i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 - 0.755i)T \)
3 \( 1 + (0.841 - 0.540i)T \)
5 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-1.68 - 4.49i)T \)
good7 \( 1 + (-0.170 + 0.0501i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (0.0281 + 0.0324i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (4.70 + 1.38i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.635 - 4.41i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.832 - 5.79i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.802 + 5.58i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (8.62 + 5.54i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (0.965 - 2.11i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (-1.37 - 3.00i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.26 - 2.74i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 9.19T + 47T^{2} \)
53 \( 1 + (10.0 - 2.94i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (-12.6 - 3.70i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (10.1 + 6.49i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (6.02 - 6.95i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-9.86 + 11.3i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (1.21 + 8.45i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-6.03 - 1.77i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (1.27 - 2.79i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (12.3 - 7.90i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (1.37 + 3.01i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.74280076940738495376635042951, −9.859049771064503200598981497578, −9.501920021952106424010331136947, −8.074871957609658615488868877773, −7.54547079154785335349307486008, −6.39101643826402103991199458748, −5.71811986792493225657723732566, −4.73867385279332970688645580620, −3.46217963615700684675888184537, −1.80645282787028108185270243876, 0.28885465152642374237095520585, 1.87547301934310689834829557665, 3.03186013473753070840855769832, 4.72252085031010407099505666753, 5.18727817307220382444738407685, 6.85869763452976235414248044492, 7.20986606847590500650987637993, 8.523594831847590901742994621869, 9.250789856829646483914171605726, 9.991119337839530432788038093908

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