Properties

Label 2-690-115.114-c2-0-47
Degree $2$
Conductor $690$
Sign $-0.495 - 0.868i$
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  − 1.41i·2-s − 1.73i·3-s − 2.00·4-s + (4.91 − 0.894i)5-s − 2.44·6-s − 6.65·7-s + 2.82i·8-s − 2.99·9-s + (−1.26 − 6.95i)10-s − 10.9i·11-s + 3.46i·12-s + 2.68i·13-s + 9.41i·14-s + (−1.54 − 8.52i)15-s + 4.00·16-s − 26.7·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.500·4-s + (0.983 − 0.178i)5-s − 0.408·6-s − 0.950·7-s + 0.353i·8-s − 0.333·9-s + (−0.126 − 0.695i)10-s − 0.998i·11-s + 0.288i·12-s + 0.206i·13-s + 0.672i·14-s + (−0.103 − 0.568i)15-s + 0.250·16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.495 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.495 - 0.868i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ -0.495 - 0.868i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4557274754\)
\(L(\frac12)\) \(\approx\) \(0.4557274754\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 + 1.73iT \)
5 \( 1 + (-4.91 + 0.894i)T \)
23 \( 1 + (7.63 + 21.6i)T \)
good7 \( 1 + 6.65T + 49T^{2} \)
11 \( 1 + 10.9iT - 121T^{2} \)
13 \( 1 - 2.68iT - 169T^{2} \)
17 \( 1 + 26.7T + 289T^{2} \)
19 \( 1 - 21.8iT - 361T^{2} \)
29 \( 1 + 24.5T + 841T^{2} \)
31 \( 1 - 19.2T + 961T^{2} \)
37 \( 1 + 22.7T + 1.36e3T^{2} \)
41 \( 1 + 66.8T + 1.68e3T^{2} \)
43 \( 1 + 41.3T + 1.84e3T^{2} \)
47 \( 1 - 3.50iT - 2.20e3T^{2} \)
53 \( 1 + 18.9T + 2.80e3T^{2} \)
59 \( 1 - 62.7T + 3.48e3T^{2} \)
61 \( 1 - 77.7iT - 3.72e3T^{2} \)
67 \( 1 - 7.04T + 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 - 111. iT - 6.24e3T^{2} \)
83 \( 1 - 13.4T + 6.88e3T^{2} \)
89 \( 1 + 55.4iT - 7.92e3T^{2} \)
97 \( 1 + 61.5T + 9.40e3T^{2} \)
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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

−9.806343791998833994435709025791, −8.870590788050630814202687773981, −8.309389316617615817722144317605, −6.71046661639973136084382033117, −6.22344213309461966236129403078, −5.18684505088006279538941017562, −3.78729336392245915551972165864, −2.67705690702188136518554823831, −1.66764650154557165065665299146, −0.15030720922982821563473859958, 2.12795681489989203016532622571, 3.42140578524163240638231283171, 4.68440144228834957668500576836, 5.44170264405237905321432913829, 6.63460573194153670435709646558, 6.90503048300779149195726740507, 8.389320790537331370765524104494, 9.376809914387626528535501620471, 9.674134360344079155002292591023, 10.53283575619174479855335186379

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