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Properties

Label 2-690-5.3-c2-0-26
Degree $2$
Conductor $690$
Sign $0.695 + 0.718i$
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 + (−2.55 − 4.29i)5-s + 2.44·6-s + (5.47 − 5.47i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.85 + 1.73i)10-s + 18.3·11-s + (−2.44 + 2.44i)12-s + (7.21 + 7.21i)13-s + 10.9i·14-s + (−2.12 + 8.39i)15-s − 4·16-s + (1.48 − 1.48i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.511 − 0.859i)5-s + 0.408·6-s + (0.781 − 0.781i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.685 + 0.173i)10-s + 1.66·11-s + (−0.204 + 0.204i)12-s + (0.555 + 0.555i)13-s + 0.781i·14-s + (−0.141 + 0.559i)15-s − 0.250·16-s + (0.0876 − 0.0876i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.695 + 0.718i$
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: \((2,\ 690,\ (\ :1),\ 0.695 + 0.718i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.379063693\)
\(L(\frac12)\) \(\approx\) \(1.379063693\)
\(L(2)\) 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 \( 1 + (2.55 + 4.29i)T \)
23 \( 1 + (3.39 + 3.39i)T \)
good7 \( 1 + (-5.47 + 5.47i)T - 49iT^{2} \)
11 \( 1 - 18.3T + 121T^{2} \)
13 \( 1 + (-7.21 - 7.21i)T + 169iT^{2} \)
17 \( 1 + (-1.48 + 1.48i)T - 289iT^{2} \)
19 \( 1 - 13.9iT - 361T^{2} \)
29 \( 1 + 0.950iT - 841T^{2} \)
31 \( 1 - 56.4T + 961T^{2} \)
37 \( 1 + (-44.4 + 44.4i)T - 1.36e3iT^{2} \)
41 \( 1 + 44.8T + 1.68e3T^{2} \)
43 \( 1 + (-31.1 - 31.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-25.7 + 25.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (17.7 + 17.7i)T + 2.80e3iT^{2} \)
59 \( 1 + 0.459iT - 3.48e3T^{2} \)
61 \( 1 - 24.6T + 3.72e3T^{2} \)
67 \( 1 + (32.6 - 32.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 46.7T + 5.04e3T^{2} \)
73 \( 1 + (0.372 + 0.372i)T + 5.32e3iT^{2} \)
79 \( 1 + 65.5iT - 6.24e3T^{2} \)
83 \( 1 + (85.4 + 85.4i)T + 6.88e3iT^{2} \)
89 \( 1 - 64.0iT - 7.92e3T^{2} \)
97 \( 1 + (-49.4 + 49.4i)T - 9.40e3iT^{2} \)
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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

−10.04267295667480857509677877962, −9.089647132413498780597282194631, −8.331106341442487270905384149052, −7.61712291948603089914127741281, −6.70076304587156086545493457316, −5.87750796182142000542692150562, −4.58315494043855628658288993886, −3.99153043005889665721539326804, −1.56231218175916672679861552439, −0.849500017079071970094686594727, 1.10084096592676450641722417644, 2.64102494809488083173053416796, 3.73477930404055251285685080714, 4.67123995220342900534523437133, 6.06248550226633955230660661014, 6.80518907212257205173563605383, 8.018560476970477109177403428271, 8.702229736289261198165809259905, 9.615554269608766508305102754123, 10.47385463064293361925126745347

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