Properties

Label 2-690-23.22-c2-0-9
Degree $2$
Conductor $690$
Sign $0.655 - 0.755i$
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 − 0.411i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s + 6.98i·11-s + 3.46·12-s − 7.65·13-s + 0.581i·14-s − 3.87i·15-s + 4.00·16-s + 22.0i·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 − 0.0587i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s + 0.635i·11-s + 0.288·12-s − 0.589·13-s + 0.0415i·14-s − 0.258i·15-s + 0.250·16-s + 1.29i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.655 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.655 - 0.755i$
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: \((2,\ 690,\ (\ :1),\ 0.655 - 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.448090529\)
\(L(\frac12)\) \(\approx\) \(1.448090529\)
\(L(2)\) 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 \( 1 + (-17.3 - 15.0i)T \)
good7 \( 1 + 0.411iT - 49T^{2} \)
11 \( 1 - 6.98iT - 121T^{2} \)
13 \( 1 + 7.65T + 169T^{2} \)
17 \( 1 - 22.0iT - 289T^{2} \)
19 \( 1 - 8.31iT - 361T^{2} \)
29 \( 1 - 32.6T + 841T^{2} \)
31 \( 1 + 17.7T + 961T^{2} \)
37 \( 1 - 22.9iT - 1.36e3T^{2} \)
41 \( 1 - 28.8T + 1.68e3T^{2} \)
43 \( 1 - 7.58iT - 1.84e3T^{2} \)
47 \( 1 - 49.4T + 2.20e3T^{2} \)
53 \( 1 - 17.5iT - 2.80e3T^{2} \)
59 \( 1 + 33.8T + 3.48e3T^{2} \)
61 \( 1 + 39.2iT - 3.72e3T^{2} \)
67 \( 1 + 70.8iT - 4.48e3T^{2} \)
71 \( 1 - 26.8T + 5.04e3T^{2} \)
73 \( 1 + 62.0T + 5.32e3T^{2} \)
79 \( 1 - 35.2iT - 6.24e3T^{2} \)
83 \( 1 - 143. iT - 6.88e3T^{2} \)
89 \( 1 - 73.4iT - 7.92e3T^{2} \)
97 \( 1 + 95.9iT - 9.40e3T^{2} \)
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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

−10.18820926788673967262615954060, −9.455933734993917024466351386850, −8.699744979094080627904223923120, −7.892499141199764802632972154688, −7.17090933616893153710117208471, −6.07586362525930550603050660112, −4.84954369496930069096492014827, −3.72420796735117833540506618052, −2.39387528891924465341535729068, −1.25672705765105750023486853204, 0.66627820589697966366267889349, 2.40496655286714357002021076956, 3.10786763013982257052488058609, 4.56447247872499075288375693069, 5.81375558853240192151985707316, 7.01822629536105277231050631339, 7.45135875512848624668168877301, 8.635732780643170506920027125030, 9.141054351007107091305944228738, 10.07684116356155498417120685696

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