Properties

Label 2-5290-1.1-c1-0-75
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 − 2.14·3-s + 4-s − 5-s + 2.14·6-s + 3.62·7-s − 8-s + 1.59·9-s + 10-s − 5.65·11-s − 2.14·12-s − 0.701·13-s − 3.62·14-s + 2.14·15-s + 16-s + 5.75·17-s − 1.59·18-s − 2.95·19-s − 20-s − 7.76·21-s + 5.65·22-s + 2.14·24-s + 25-s + 0.701·26-s + 3.01·27-s + 3.62·28-s + 0.108·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.23·3-s + 0.5·4-s − 0.447·5-s + 0.874·6-s + 1.36·7-s − 0.353·8-s + 0.531·9-s + 0.316·10-s − 1.70·11-s − 0.618·12-s − 0.194·13-s − 0.968·14-s + 0.553·15-s + 0.250·16-s + 1.39·17-s − 0.375·18-s − 0.677·19-s − 0.223·20-s − 1.69·21-s + 1.20·22-s + 0.437·24-s + 0.200·25-s + 0.137·26-s + 0.580·27-s + 0.684·28-s + 0.0200·29-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: Trivial
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 \( 1 + 2.14T + 3T^{2} \)
7 \( 1 - 3.62T + 7T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 + 0.701T + 13T^{2} \)
17 \( 1 - 5.75T + 17T^{2} \)
19 \( 1 + 2.95T + 19T^{2} \)
29 \( 1 - 0.108T + 29T^{2} \)
31 \( 1 + 2.03T + 31T^{2} \)
37 \( 1 + 2.92T + 37T^{2} \)
41 \( 1 - 8.21T + 41T^{2} \)
43 \( 1 - 7.57T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 6.83T + 53T^{2} \)
59 \( 1 + 5.41T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 - 9.07T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 2.95T + 89T^{2} \)
97 \( 1 - 0.0397T + 97T^{2} \)
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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

−7.943486436955566843305640024381, −7.36678675577389299568147984206, −6.37493488101529442026568839225, −5.53207660605296466430230196917, −5.14383500293286863944501830605, −4.42151871124490510043247546866, −3.12992959547042982636463054979, −2.12168149499530784560320884244, −1.02553295340857869141668872611, 0, 1.02553295340857869141668872611, 2.12168149499530784560320884244, 3.12992959547042982636463054979, 4.42151871124490510043247546866, 5.14383500293286863944501830605, 5.53207660605296466430230196917, 6.37493488101529442026568839225, 7.36678675577389299568147984206, 7.943486436955566843305640024381

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