Properties

Label 2-690-1.1-c1-0-7
Degree $2$
Conductor $690$
Sign $1$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 2·11-s + 12-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s + 2·22-s + 23-s + 24-s + 25-s + 27-s − 30-s − 8·31-s + 32-s + 2·33-s + 6·34-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s + 0.288·12-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.426·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.182·30-s − 1.43·31-s + 0.176·32-s + 0.348·33-s + 1.02·34-s + 1/6·36-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{690} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.768026046\)
\(L(\frac12)\) \(\approx\) \(2.768026046\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 + T \)
23 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−10.50414721591216363016256238125, −9.617810723188203482804610348135, −8.704053618561596117018599927298, −7.65887835624656816954704262462, −7.09826948100239356123949864369, −5.86867619303862171783314444659, −4.91958962439090979700956862591, −3.73046488723343456020472648480, −3.11980143916399740040225485646, −1.51282503790287704249684760774, 1.51282503790287704249684760774, 3.11980143916399740040225485646, 3.73046488723343456020472648480, 4.91958962439090979700956862591, 5.86867619303862171783314444659, 7.09826948100239356123949864369, 7.65887835624656816954704262462, 8.704053618561596117018599927298, 9.617810723188203482804610348135, 10.50414721591216363016256238125

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