Properties

Label 2-7600-1.1-c1-0-166
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  + 2.79·3-s + 0.127·7-s + 4.83·9-s − 5.21·11-s + 0.515·13-s − 3.58·17-s − 19-s + 0.357·21-s − 6.50·23-s + 5.14·27-s − 3.05·29-s − 5.44·31-s − 14.5·33-s + 4.99·37-s + 1.44·39-s + 11.4·41-s − 2.06·43-s + 11.9·47-s − 6.98·49-s − 10.0·51-s − 2.25·53-s − 2.79·57-s − 1.89·59-s − 5.83·61-s + 0.616·63-s + 0.432·67-s − 18.2·69-s + ⋯
L(s)  = 1  + 1.61·3-s + 0.0482·7-s + 1.61·9-s − 1.57·11-s + 0.142·13-s − 0.868·17-s − 0.229·19-s + 0.0779·21-s − 1.35·23-s + 0.989·27-s − 0.567·29-s − 0.977·31-s − 2.54·33-s + 0.821·37-s + 0.231·39-s + 1.78·41-s − 0.315·43-s + 1.73·47-s − 0.997·49-s − 1.40·51-s − 0.310·53-s − 0.370·57-s − 0.246·59-s − 0.746·61-s + 0.0777·63-s + 0.0528·67-s − 2.19·69-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 \( 1 - 2.79T + 3T^{2} \)
7 \( 1 - 0.127T + 7T^{2} \)
11 \( 1 + 5.21T + 11T^{2} \)
13 \( 1 - 0.515T + 13T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
23 \( 1 + 6.50T + 23T^{2} \)
29 \( 1 + 3.05T + 29T^{2} \)
31 \( 1 + 5.44T + 31T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 2.06T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 2.25T + 53T^{2} \)
59 \( 1 + 1.89T + 59T^{2} \)
61 \( 1 + 5.83T + 61T^{2} \)
67 \( 1 - 0.432T + 67T^{2} \)
71 \( 1 + 4.77T + 71T^{2} \)
73 \( 1 + 9.98T + 73T^{2} \)
79 \( 1 - 3.28T + 79T^{2} \)
83 \( 1 - 0.496T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 7.00T + 97T^{2} \)
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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

−7.74731768882923824968085873356, −7.19910253217697571797938679502, −6.13312629026799311666424153267, −5.43731009308568370301936029729, −4.36676476902041552046454200249, −3.93676183210762841659172416940, −2.87159552950521352379257621528, −2.45372036117674958744443515264, −1.67391833147898922553787888617, 0, 1.67391833147898922553787888617, 2.45372036117674958744443515264, 2.87159552950521352379257621528, 3.93676183210762841659172416940, 4.36676476902041552046454200249, 5.43731009308568370301936029729, 6.13312629026799311666424153267, 7.19910253217697571797938679502, 7.74731768882923824968085873356

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