Properties

Label 2-7600-1.1-c1-0-0
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.296·3-s − 3.56·7-s − 2.91·9-s − 5.56·11-s − 5.26·13-s − 1.40·17-s − 19-s − 1.05·21-s − 6.96·23-s − 1.75·27-s + 1.40·29-s − 1.75·31-s − 1.65·33-s − 3.61·37-s − 1.56·39-s + 4.34·41-s − 3.56·43-s − 8.26·47-s + 5.69·49-s − 0.417·51-s + 7.61·53-s − 0.296·57-s − 9.47·59-s + 9.21·61-s + 10.3·63-s − 4.76·67-s − 2.06·69-s + ⋯
L(s)  = 1  + 0.171·3-s − 1.34·7-s − 0.970·9-s − 1.67·11-s − 1.46·13-s − 0.341·17-s − 0.229·19-s − 0.230·21-s − 1.45·23-s − 0.337·27-s + 0.261·29-s − 0.315·31-s − 0.287·33-s − 0.594·37-s − 0.250·39-s + 0.679·41-s − 0.543·43-s − 1.20·47-s + 0.813·49-s − 0.0584·51-s + 1.04·53-s − 0.0393·57-s − 1.23·59-s + 1.17·61-s + 1.30·63-s − 0.581·67-s − 0.249·69-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: \(0\)
Selberg data: \((2,\ 7600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04162749063\)
\(L(\frac12)\) \(\approx\) \(0.04162749063\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.296T + 3T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
13 \( 1 + 5.26T + 13T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
23 \( 1 + 6.96T + 23T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 + 1.75T + 31T^{2} \)
37 \( 1 + 3.61T + 37T^{2} \)
41 \( 1 - 4.34T + 41T^{2} \)
43 \( 1 + 3.56T + 43T^{2} \)
47 \( 1 + 8.26T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 + 9.47T + 59T^{2} \)
61 \( 1 - 9.21T + 61T^{2} \)
67 \( 1 + 4.76T + 67T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + 6.59T + 73T^{2} \)
79 \( 1 + 5.47T + 79T^{2} \)
83 \( 1 + 4.15T + 83T^{2} \)
89 \( 1 + 9.23T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.948575053878578473162706046421, −7.19246870737027069139760060391, −6.51386762524013540192166160026, −5.70333046638378925679720093228, −5.23792128490208083488630031600, −4.30921970568964373715797949039, −3.30664647086535217597160103912, −2.68955139490863659365744285294, −2.16235674291035441962226381216, −0.090416844964688394101384681719, 0.090416844964688394101384681719, 2.16235674291035441962226381216, 2.68955139490863659365744285294, 3.30664647086535217597160103912, 4.30921970568964373715797949039, 5.23792128490208083488630031600, 5.70333046638378925679720093228, 6.51386762524013540192166160026, 7.19246870737027069139760060391, 7.948575053878578473162706046421

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