Properties

Label 2-690-23.22-c2-0-6
Degree $2$
Conductor $690$
Sign $0.211 - 0.977i$
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  + 1.41·2-s − 1.73·3-s + 2.00·4-s − 2.23i·5-s − 2.44·6-s + 1.37i·7-s + 2.82·8-s + 2.99·9-s − 3.16i·10-s + 12.6i·11-s − 3.46·12-s − 17.1·13-s + 1.93i·14-s + 3.87i·15-s + 4.00·16-s + 17.1i·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.500·4-s − 0.447i·5-s − 0.408·6-s + 0.195i·7-s + 0.353·8-s + 0.333·9-s − 0.316i·10-s + 1.14i·11-s − 0.288·12-s − 1.32·13-s + 0.138i·14-s + 0.258i·15-s + 0.250·16-s + 1.01i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.211 - 0.977i$
Analytic conductor: \(18.8011\)
Root analytic conductor: \(4.33602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1),\ 0.211 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.838471296\)
\(L(\frac12)\) \(\approx\) \(1.838471296\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 + 1.73T \)
5 \( 1 + 2.23iT \)
23 \( 1 + (-22.4 - 4.85i)T \)
good7 \( 1 - 1.37iT - 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 + 17.1T + 169T^{2} \)
17 \( 1 - 17.1iT - 289T^{2} \)
19 \( 1 - 5.73iT - 361T^{2} \)
29 \( 1 - 19.0T + 841T^{2} \)
31 \( 1 - 27.6T + 961T^{2} \)
37 \( 1 - 28.5iT - 1.36e3T^{2} \)
41 \( 1 + 40.8T + 1.68e3T^{2} \)
43 \( 1 - 33.6iT - 1.84e3T^{2} \)
47 \( 1 + 36.0T + 2.20e3T^{2} \)
53 \( 1 - 72.1iT - 2.80e3T^{2} \)
59 \( 1 - 33.0T + 3.48e3T^{2} \)
61 \( 1 - 102. iT - 3.72e3T^{2} \)
67 \( 1 - 83.7iT - 4.48e3T^{2} \)
71 \( 1 + 53.3T + 5.04e3T^{2} \)
73 \( 1 + 6.09T + 5.32e3T^{2} \)
79 \( 1 + 16.0iT - 6.24e3T^{2} \)
83 \( 1 + 119. iT - 6.88e3T^{2} \)
89 \( 1 + 71.6iT - 7.92e3T^{2} \)
97 \( 1 + 47.4iT - 9.40e3T^{2} \)
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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

−10.34145472871227226210009032883, −9.931017154115471619023241645191, −8.709916614058026779810957408950, −7.57837327233663524430560047362, −6.82912230582467131774338258259, −5.82119931058437307363257649622, −4.86301510735332940481210362294, −4.33983539829354429754332324695, −2.77771063494174823838579245773, −1.47943384772277361956431096277, 0.56193342436362152182539466313, 2.46953388653665432135062114825, 3.43796219524850813596571454486, 4.77849323686033224645890866090, 5.37581521469734244875299265302, 6.57638697822208245709184040665, 7.07770759535780162919553632767, 8.163506878241468704354952924496, 9.410270325316784559393628417911, 10.32214041998222785266011692775

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