Properties

Label 2-7600-1.1-c1-0-108
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  − 2.77·3-s + 4.69·7-s + 4.71·9-s − 6.40·11-s + 1.06·13-s + 1.91·17-s + 19-s − 13.0·21-s − 1.79·23-s − 4.75·27-s + 2.93·29-s + 5.55·31-s + 17.7·33-s − 11.4·37-s − 2.95·39-s − 1.14·41-s + 3.55·43-s − 10.8·47-s + 15.0·49-s − 5.32·51-s − 8.69·53-s − 2.77·57-s + 5.63·59-s − 3.39·61-s + 22.1·63-s + 8.82·67-s + 4.98·69-s + ⋯
L(s)  = 1  − 1.60·3-s + 1.77·7-s + 1.57·9-s − 1.93·11-s + 0.295·13-s + 0.465·17-s + 0.229·19-s − 2.84·21-s − 0.374·23-s − 0.915·27-s + 0.545·29-s + 0.997·31-s + 3.09·33-s − 1.87·37-s − 0.473·39-s − 0.178·41-s + 0.542·43-s − 1.58·47-s + 2.14·49-s − 0.745·51-s − 1.19·53-s − 0.367·57-s + 0.733·59-s − 0.434·61-s + 2.78·63-s + 1.07·67-s + 0.600·69-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: $\chi_{7600} (1, \cdot )$
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 \( 1 + 2.77T + 3T^{2} \)
7 \( 1 - 4.69T + 7T^{2} \)
11 \( 1 + 6.40T + 11T^{2} \)
13 \( 1 - 1.06T + 13T^{2} \)
17 \( 1 - 1.91T + 17T^{2} \)
23 \( 1 + 1.79T + 23T^{2} \)
29 \( 1 - 2.93T + 29T^{2} \)
31 \( 1 - 5.55T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 + 1.14T + 41T^{2} \)
43 \( 1 - 3.55T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 8.69T + 53T^{2} \)
59 \( 1 - 5.63T + 59T^{2} \)
61 \( 1 + 3.39T + 61T^{2} \)
67 \( 1 - 8.82T + 67T^{2} \)
71 \( 1 - 1.42T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 1.96T + 79T^{2} \)
83 \( 1 + 16.2T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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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

−7.52694721425662980724354414992, −6.81588076599967759194689562028, −5.85811662851070170973867112904, −5.36874853555755854591952426657, −4.90013810607829543385536330002, −4.41471614223093769743007724274, −3.09241336888306745811816933900, −1.97337521072460976665496687686, −1.13708898012126983316434908206, 0, 1.13708898012126983316434908206, 1.97337521072460976665496687686, 3.09241336888306745811816933900, 4.41471614223093769743007724274, 4.90013810607829543385536330002, 5.36874853555755854591952426657, 5.85811662851070170973867112904, 6.81588076599967759194689562028, 7.52694721425662980724354414992

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