Properties

Label 2-7600-1.1-c1-0-119
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  − 1.70·3-s + 1.07·7-s − 0.0783·9-s + 6.34·11-s − 1.36·13-s − 3.26·17-s + 19-s − 1.84·21-s + 2.34·23-s + 5.26·27-s + 1.41·29-s − 8.68·31-s − 10.8·33-s − 5.36·37-s + 2.34·39-s − 3.26·41-s − 11.9·43-s + 1.07·47-s − 5.83·49-s + 5.57·51-s − 6.63·53-s − 1.70·57-s + 11.4·59-s + 5.60·61-s − 0.0845·63-s + 10.3·67-s − 3.99·69-s + ⋯
L(s)  = 1  − 0.986·3-s + 0.407·7-s − 0.0261·9-s + 1.91·11-s − 0.379·13-s − 0.791·17-s + 0.229·19-s − 0.402·21-s + 0.487·23-s + 1.01·27-s + 0.263·29-s − 1.55·31-s − 1.88·33-s − 0.882·37-s + 0.374·39-s − 0.509·41-s − 1.81·43-s + 0.157·47-s − 0.833·49-s + 0.780·51-s − 0.910·53-s − 0.226·57-s + 1.48·59-s + 0.717·61-s − 0.0106·63-s + 1.26·67-s − 0.481·69-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 \( 1 + 1.70T + 3T^{2} \)
7 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 - 6.34T + 11T^{2} \)
13 \( 1 + 1.36T + 13T^{2} \)
17 \( 1 + 3.26T + 17T^{2} \)
23 \( 1 - 2.34T + 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + 5.36T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 1.07T + 47T^{2} \)
53 \( 1 + 6.63T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 - 5.60T + 61T^{2} \)
67 \( 1 - 10.3T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 5.41T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 - 7.57T + 89T^{2} \)
97 \( 1 - 8.88T + 97T^{2} \)
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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

−7.20587034560771748946361471141, −6.76374503752299927766229964076, −6.25286391815367029348990602654, −5.32471567192947358463819477022, −4.89844708967095191010746352855, −4.00318498725935834869856463843, −3.29329174240933181629753615829, −2.00348846800412125682859589359, −1.20741229090836664048535132932, 0, 1.20741229090836664048535132932, 2.00348846800412125682859589359, 3.29329174240933181629753615829, 4.00318498725935834869856463843, 4.89844708967095191010746352855, 5.32471567192947358463819477022, 6.25286391815367029348990602654, 6.76374503752299927766229964076, 7.20587034560771748946361471141

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