Properties

Label 2-7600-1.1-c1-0-98
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  − 3.25·3-s + 0.0778·7-s + 7.58·9-s + 4.50·11-s − 5.33·13-s + 7.33·17-s − 19-s − 0.253·21-s − 3.40·23-s − 14.9·27-s − 1.33·29-s + 2.50·31-s − 14.6·33-s − 5.50·37-s + 17.3·39-s − 0.506·43-s + 5.66·47-s − 6.99·49-s − 23.8·51-s − 12.9·53-s + 3.25·57-s − 7.56·59-s − 2.15·61-s + 0.590·63-s + 4.58·67-s + 11.0·69-s + 10.8·71-s + ⋯
L(s)  = 1  − 1.87·3-s + 0.0294·7-s + 2.52·9-s + 1.35·11-s − 1.47·13-s + 1.77·17-s − 0.229·19-s − 0.0553·21-s − 0.710·23-s − 2.87·27-s − 0.247·29-s + 0.450·31-s − 2.55·33-s − 0.905·37-s + 2.77·39-s − 0.0772·43-s + 0.825·47-s − 0.999·49-s − 3.33·51-s − 1.77·53-s + 0.430·57-s − 0.984·59-s − 0.276·61-s + 0.0744·63-s + 0.560·67-s + 1.33·69-s + 1.28·71-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: $\chi_{7600} (1, \cdot )$
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 \( 1 + 3.25T + 3T^{2} \)
7 \( 1 - 0.0778T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
13 \( 1 + 5.33T + 13T^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
23 \( 1 + 3.40T + 23T^{2} \)
29 \( 1 + 1.33T + 29T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + 5.50T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 0.506T + 43T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 7.56T + 59T^{2} \)
61 \( 1 + 2.15T + 61T^{2} \)
67 \( 1 - 4.58T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 5.09T + 73T^{2} \)
79 \( 1 + 17.0T + 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−7.43252641688931278306949354829, −6.58144074292969707098529224222, −6.21052254716457205903760724630, −5.40380181716028184326246896329, −4.89280247241405012816042875204, −4.18523669198341327892356789079, −3.33477135401643636310551851358, −1.86777102014407876208686090581, −1.07490997775637325665094544257, 0, 1.07490997775637325665094544257, 1.86777102014407876208686090581, 3.33477135401643636310551851358, 4.18523669198341327892356789079, 4.89280247241405012816042875204, 5.40380181716028184326246896329, 6.21052254716457205903760724630, 6.58144074292969707098529224222, 7.43252641688931278306949354829

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