Properties

Label 2-7600-1.1-c1-0-130
Degree $2$
Conductor $7600$
Sign $-1$
Analytic cond. $60.6863$
Root an. cond. $7.79014$
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  + 0.471·3-s + 0.567·7-s − 2.77·9-s − 4.37·11-s + 0.165·13-s + 7.94·17-s − 19-s + 0.267·21-s + 3.87·23-s − 2.72·27-s + 3.53·29-s − 3.20·31-s − 2.06·33-s − 10.1·37-s + 0.0779·39-s − 5.97·41-s + 12.0·43-s − 5.46·47-s − 6.67·49-s + 3.74·51-s + 2.00·53-s − 0.471·57-s − 8.32·59-s + 11.8·61-s − 1.57·63-s + 8.79·67-s + 1.82·69-s + ⋯
L(s)  = 1  + 0.271·3-s + 0.214·7-s − 0.926·9-s − 1.32·11-s + 0.0458·13-s + 1.92·17-s − 0.229·19-s + 0.0583·21-s + 0.807·23-s − 0.523·27-s + 0.655·29-s − 0.575·31-s − 0.358·33-s − 1.67·37-s + 0.0124·39-s − 0.932·41-s + 1.84·43-s − 0.796·47-s − 0.954·49-s + 0.524·51-s + 0.275·53-s − 0.0623·57-s − 1.08·59-s + 1.51·61-s − 0.198·63-s + 1.07·67-s + 0.219·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7600\)    =    \(2^{4} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(60.6863\)
Root analytic conductor: \(7.79014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 0.471T + 3T^{2} \)
7 \( 1 - 0.567T + 7T^{2} \)
11 \( 1 + 4.37T + 11T^{2} \)
13 \( 1 - 0.165T + 13T^{2} \)
17 \( 1 - 7.94T + 17T^{2} \)
23 \( 1 - 3.87T + 23T^{2} \)
29 \( 1 - 3.53T + 29T^{2} \)
31 \( 1 + 3.20T + 31T^{2} \)
37 \( 1 + 10.1T + 37T^{2} \)
41 \( 1 + 5.97T + 41T^{2} \)
43 \( 1 - 12.0T + 43T^{2} \)
47 \( 1 + 5.46T + 47T^{2} \)
53 \( 1 - 2.00T + 53T^{2} \)
59 \( 1 + 8.32T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 8.79T + 67T^{2} \)
71 \( 1 + 0.720T + 71T^{2} \)
73 \( 1 + 4.54T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + 6.72T + 83T^{2} \)
89 \( 1 - 8.11T + 89T^{2} \)
97 \( 1 - 13.6T + 97T^{2} \)
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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

−7.68909317326917668645600076380, −6.97047579900342943959762309364, −5.99071715937105998111442160912, −5.31167279611936087951353443151, −4.99857297745945653659574786172, −3.70110372760700890770195139898, −3.09205691665947725491564995582, −2.42255912755887699568598329853, −1.28045491613866060849014307113, 0, 1.28045491613866060849014307113, 2.42255912755887699568598329853, 3.09205691665947725491564995582, 3.70110372760700890770195139898, 4.99857297745945653659574786172, 5.31167279611936087951353443151, 5.99071715937105998111442160912, 6.97047579900342943959762309364, 7.68909317326917668645600076380

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