Properties

Label 2-760-1.1-c1-0-13
Degree $2$
Conductor $760$
Sign $-1$
Analytic cond. $6.06863$
Root an. cond. $2.46345$
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  − 0.470·3-s − 5-s + 2.71·7-s − 2.77·9-s − 5.55·11-s + 2.02·13-s + 0.470·15-s − 3.77·17-s + 19-s − 1.28·21-s − 5.77·23-s + 25-s + 2.71·27-s − 5.66·29-s + 7.55·31-s + 2.61·33-s − 2.71·35-s − 3.75·37-s − 0.954·39-s − 12.6·41-s − 9.43·43-s + 2.77·45-s + 11.1·47-s + 0.397·49-s + 1.77·51-s − 8.85·53-s + 5.55·55-s + ⋯
L(s)  = 1  − 0.271·3-s − 0.447·5-s + 1.02·7-s − 0.926·9-s − 1.67·11-s + 0.562·13-s + 0.121·15-s − 0.916·17-s + 0.229·19-s − 0.279·21-s − 1.20·23-s + 0.200·25-s + 0.523·27-s − 1.05·29-s + 1.35·31-s + 0.455·33-s − 0.459·35-s − 0.616·37-s − 0.152·39-s − 1.97·41-s − 1.43·43-s + 0.414·45-s + 1.62·47-s + 0.0567·49-s + 0.249·51-s − 1.21·53-s + 0.749·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-1$
Analytic conductor: \(6.06863\)
Root analytic conductor: \(2.46345\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 760,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
19 \( 1 - T \)
good3 \( 1 + 0.470T + 3T^{2} \)
7 \( 1 - 2.71T + 7T^{2} \)
11 \( 1 + 5.55T + 11T^{2} \)
13 \( 1 - 2.02T + 13T^{2} \)
17 \( 1 + 3.77T + 17T^{2} \)
23 \( 1 + 5.77T + 23T^{2} \)
29 \( 1 + 5.66T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 + 3.75T + 37T^{2} \)
41 \( 1 + 12.6T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 - 11.1T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.45T + 73T^{2} \)
79 \( 1 - 8.94T + 79T^{2} \)
83 \( 1 + 4.94T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.20020928669431709241909838779, −8.737575313337994366301470568337, −8.228861095782982207773762068028, −7.53096182092576360959363346625, −6.26440952286459499269251298369, −5.31852128526798038013231331601, −4.62080971355182172365566202781, −3.25415039825249590894571527107, −2.02150641146546994728047556882, 0, 2.02150641146546994728047556882, 3.25415039825249590894571527107, 4.62080971355182172365566202781, 5.31852128526798038013231331601, 6.26440952286459499269251298369, 7.53096182092576360959363346625, 8.228861095782982207773762068028, 8.737575313337994366301470568337, 10.20020928669431709241909838779

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