Properties

Label 2-690-23.22-c2-0-18
Degree $2$
Conductor $690$
Sign $0.141 + 0.989i$
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 − 8.87i·7-s − 2.82·8-s + 2.99·9-s + 3.16i·10-s + 6.26i·11-s − 3.46·12-s + 22.5·13-s + 12.5i·14-s + 3.87i·15-s + 4.00·16-s + 9.43i·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 − 1.26i·7-s − 0.353·8-s + 0.333·9-s + 0.316i·10-s + 0.569i·11-s − 0.288·12-s + 1.73·13-s + 0.896i·14-s + 0.258i·15-s + 0.250·16-s + 0.555i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.141 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.141 + 0.989i$
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: \((2,\ 690,\ (\ :1),\ 0.141 + 0.989i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.054793336\)
\(L(\frac12)\) \(\approx\) \(1.054793336\)
\(L(2)\) 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 \( 1 + (-22.7 + 3.26i)T \)
good7 \( 1 + 8.87iT - 49T^{2} \)
11 \( 1 - 6.26iT - 121T^{2} \)
13 \( 1 - 22.5T + 169T^{2} \)
17 \( 1 - 9.43iT - 289T^{2} \)
19 \( 1 + 10.5iT - 361T^{2} \)
29 \( 1 + 7.04T + 841T^{2} \)
31 \( 1 + 18.9T + 961T^{2} \)
37 \( 1 - 39.6iT - 1.36e3T^{2} \)
41 \( 1 + 2.96T + 1.68e3T^{2} \)
43 \( 1 + 73.5iT - 1.84e3T^{2} \)
47 \( 1 - 64.8T + 2.20e3T^{2} \)
53 \( 1 - 14.0iT - 2.80e3T^{2} \)
59 \( 1 - 19.3T + 3.48e3T^{2} \)
61 \( 1 + 55.1iT - 3.72e3T^{2} \)
67 \( 1 + 41.0iT - 4.48e3T^{2} \)
71 \( 1 + 33.4T + 5.04e3T^{2} \)
73 \( 1 - 40.7T + 5.32e3T^{2} \)
79 \( 1 + 49.8iT - 6.24e3T^{2} \)
83 \( 1 + 36.6iT - 6.88e3T^{2} \)
89 \( 1 + 129. iT - 7.92e3T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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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

−10.26966752240322655800640516216, −9.173075599862444118546025091559, −8.437152602917169515553322260058, −7.37144705748309816343362648210, −6.72490727855329873677083823326, −5.70854811483569625182170997968, −4.48898961392553162968074976499, −3.54739735843656693996799799103, −1.59647551000355018259375791009, −0.63214457974345811345916993812, 1.15533518522722574332720516084, 2.59279503845630645533702098544, 3.74566205697404643491905295939, 5.46858262310932217207272728977, 5.98131901417909254538632411922, 6.87853807160162589018366454986, 7.999900783692044794666482343102, 8.856632086283408985142542614987, 9.434836261687970426110405777881, 10.68707247246859718947471797425

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