Properties

Label 2-7600-1.1-c1-0-159
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  + 1.41·3-s + 4.82·7-s − 0.999·9-s − 4.82·11-s − 0.585·13-s − 2.82·17-s + 19-s + 6.82·21-s − 7.65·23-s − 5.65·27-s + 3.65·29-s − 6.82·31-s − 6.82·33-s + 0.585·37-s − 0.828·39-s + 0.828·41-s + 8.82·43-s − 0.828·47-s + 16.3·49-s − 4.00·51-s − 11.8·53-s + 1.41·57-s + 6.82·59-s − 5.65·61-s − 4.82·63-s − 9.89·67-s − 10.8·69-s + ⋯
L(s)  = 1  + 0.816·3-s + 1.82·7-s − 0.333·9-s − 1.45·11-s − 0.162·13-s − 0.685·17-s + 0.229·19-s + 1.49·21-s − 1.59·23-s − 1.08·27-s + 0.679·29-s − 1.22·31-s − 1.18·33-s + 0.0963·37-s − 0.132·39-s + 0.129·41-s + 1.34·43-s − 0.120·47-s + 2.33·49-s − 0.560·51-s − 1.63·53-s + 0.187·57-s + 0.888·59-s − 0.724·61-s − 0.608·63-s − 1.20·67-s − 1.30·69-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 \( 1 - 1.41T + 3T^{2} \)
7 \( 1 - 4.82T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 0.585T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 - 0.585T + 37T^{2} \)
41 \( 1 - 0.828T + 41T^{2} \)
43 \( 1 - 8.82T + 43T^{2} \)
47 \( 1 + 0.828T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 + 5.65T + 61T^{2} \)
67 \( 1 + 9.89T + 67T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + 1.17T + 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 13.3T + 89T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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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

−7.76526701851580408560657823519, −7.19796501756956204255477343853, −5.89790640751262423856225591713, −5.42806811349004217501668103395, −4.63149370091492726338290541486, −4.03957364298085865294994701733, −2.88370797060470358113630128318, −2.28417353927317556462842344667, −1.59640012296732022880350957643, 0, 1.59640012296732022880350957643, 2.28417353927317556462842344667, 2.88370797060470358113630128318, 4.03957364298085865294994701733, 4.63149370091492726338290541486, 5.42806811349004217501668103395, 5.89790640751262423856225591713, 7.19796501756956204255477343853, 7.76526701851580408560657823519

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