Properties

Label 2-5290-1.1-c1-0-106
Degree $2$
Conductor $5290$
Sign $-1$
Analytic cond. $42.2408$
Root an. cond. $6.49929$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.79·3-s + 4-s + 5-s + 2.79·6-s + 1.79·7-s − 8-s + 4.79·9-s − 10-s + 0.791·11-s − 2.79·12-s + 5.79·13-s − 1.79·14-s − 2.79·15-s + 16-s − 0.791·17-s − 4.79·18-s − 5.79·19-s + 20-s − 5·21-s − 0.791·22-s + 2.79·24-s + 25-s − 5.79·26-s − 4.99·27-s + 1.79·28-s + 7.58·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.61·3-s + 0.5·4-s + 0.447·5-s + 1.13·6-s + 0.677·7-s − 0.353·8-s + 1.59·9-s − 0.316·10-s + 0.238·11-s − 0.805·12-s + 1.60·13-s − 0.478·14-s − 0.720·15-s + 0.250·16-s − 0.191·17-s − 1.12·18-s − 1.32·19-s + 0.223·20-s − 1.09·21-s − 0.168·22-s + 0.569·24-s + 0.200·25-s − 1.13·26-s − 0.962·27-s + 0.338·28-s + 1.40·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5290\)    =    \(2 \cdot 5 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(42.2408\)
Root analytic conductor: \(6.49929\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5290} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5290,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 2.79T + 3T^{2} \)
7 \( 1 - 1.79T + 7T^{2} \)
11 \( 1 - 0.791T + 11T^{2} \)
13 \( 1 - 5.79T + 13T^{2} \)
17 \( 1 + 0.791T + 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
29 \( 1 - 7.58T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 + 4.41T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 7.95T + 97T^{2} \)
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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

−7.983632026711558776898517608607, −6.77682055724084887237329671925, −6.39207443516318198282613810262, −5.95634571640188433173166948228, −4.98346214175879970956748007345, −4.43479227425195452680727738060, −3.26541588307983695276272334365, −1.78526885550021860631648280189, −1.25733118263330262111971044933, 0, 1.25733118263330262111971044933, 1.78526885550021860631648280189, 3.26541588307983695276272334365, 4.43479227425195452680727738060, 4.98346214175879970956748007345, 5.95634571640188433173166948228, 6.39207443516318198282613810262, 6.77682055724084887237329671925, 7.983632026711558776898517608607

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