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Properties

Label 2-690-5.2-c2-0-16
Degree $2$
Conductor $690$
Sign $0.554 + 0.832i$
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 + (−3.65 − 3.41i)5-s + 2.44·6-s + (0.283 + 0.283i)7-s + (2 − 2i)8-s − 2.99i·9-s + (0.240 + 7.06i)10-s + 8.31·11-s + (−2.44 − 2.44i)12-s + (−17.3 + 17.3i)13-s − 0.567i·14-s + (8.65 − 0.294i)15-s − 4·16-s + (9.44 + 9.44i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.408 + 0.408i)3-s + 0.5i·4-s + (−0.730 − 0.682i)5-s + 0.408·6-s + (0.0405 + 0.0405i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.0240 + 0.706i)10-s + 0.756·11-s + (−0.204 − 0.204i)12-s + (−1.33 + 1.33i)13-s − 0.0405i·14-s + (0.577 − 0.0196i)15-s − 0.250·16-s + (0.555 + 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.554 + 0.832i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8245953028\)
\(L(\frac12)\) \(\approx\) \(0.8245953028\)
\(L(2)\) 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 \( 1 + (3.65 + 3.41i)T \)
23 \( 1 + (3.39 - 3.39i)T \)
good7 \( 1 + (-0.283 - 0.283i)T + 49iT^{2} \)
11 \( 1 - 8.31T + 121T^{2} \)
13 \( 1 + (17.3 - 17.3i)T - 169iT^{2} \)
17 \( 1 + (-9.44 - 9.44i)T + 289iT^{2} \)
19 \( 1 + 24.8iT - 361T^{2} \)
29 \( 1 - 7.76iT - 841T^{2} \)
31 \( 1 + 21.7T + 961T^{2} \)
37 \( 1 + (22.4 + 22.4i)T + 1.36e3iT^{2} \)
41 \( 1 + 2.70T + 1.68e3T^{2} \)
43 \( 1 + (-51.2 + 51.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-57.3 - 57.3i)T + 2.20e3iT^{2} \)
53 \( 1 + (-57.8 + 57.8i)T - 2.80e3iT^{2} \)
59 \( 1 + 17.1iT - 3.48e3T^{2} \)
61 \( 1 - 88.9T + 3.72e3T^{2} \)
67 \( 1 + (-53.5 - 53.5i)T + 4.48e3iT^{2} \)
71 \( 1 + 43.1T + 5.04e3T^{2} \)
73 \( 1 + (-18.0 + 18.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 155. iT - 6.24e3T^{2} \)
83 \( 1 + (-69.5 + 69.5i)T - 6.88e3iT^{2} \)
89 \( 1 + 67.8iT - 7.92e3T^{2} \)
97 \( 1 + (-48.4 - 48.4i)T + 9.40e3iT^{2} \)
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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

−10.09286737125713930015904143207, −9.114624571405896769651806880388, −8.875699440954954760496971764213, −7.48737535771854544329844913720, −6.87511197456858667804037064501, −5.36822759354321494418369991527, −4.42431845914053623927047006500, −3.69060321667115279820023866737, −2.05449990700540904116057676682, −0.54803399499215383380798164539, 0.817702006288627141641841306227, 2.55711098607520634923101250987, 3.88769721563682566668422682867, 5.22660916374461068640228553843, 6.07421422015363385549192190273, 7.15603898575076684916727071974, 7.59855064012668070693575695057, 8.373208581633161871071534933151, 9.687863250688417155438357089756, 10.31708504917260341179501958847

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