Properties

Label 2-912-1.1-c1-0-13
Degree $2$
Conductor $912$
Sign $-1$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  − 3-s − 1.56·5-s + 1.56·7-s + 9-s − 5.56·11-s + 3.12·13-s + 1.56·15-s + 6.68·17-s − 19-s − 1.56·21-s − 9.12·23-s − 2.56·25-s − 27-s − 8.24·29-s + 2·31-s + 5.56·33-s − 2.43·35-s − 8·37-s − 3.12·39-s − 5.12·41-s + 1.56·43-s − 1.56·45-s + 6.68·47-s − 4.56·49-s − 6.68·51-s + 4.24·53-s + 8.68·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.698·5-s + 0.590·7-s + 0.333·9-s − 1.67·11-s + 0.866·13-s + 0.403·15-s + 1.62·17-s − 0.229·19-s − 0.340·21-s − 1.90·23-s − 0.512·25-s − 0.192·27-s − 1.53·29-s + 0.359·31-s + 0.968·33-s − 0.412·35-s − 1.31·37-s − 0.500·39-s − 0.800·41-s + 0.238·43-s − 0.232·45-s + 0.975·47-s − 0.651·49-s − 0.936·51-s + 0.583·53-s + 1.17·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-1$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
19 \( 1 + T \)
good5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 - 1.56T + 7T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
23 \( 1 + 9.12T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 - 1.56T + 43T^{2} \)
47 \( 1 - 6.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 + 6.24T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−10.04861278183804889970414996523, −8.598341296363830787311762210788, −7.83317291651819864639250492514, −7.44053253339425069190783619267, −5.90083704059303951170111712183, −5.45272936977689125253089538386, −4.29891914253423664511325975669, −3.34597243996746572161592963996, −1.77028391308079892196811307389, 0, 1.77028391308079892196811307389, 3.34597243996746572161592963996, 4.29891914253423664511325975669, 5.45272936977689125253089538386, 5.90083704059303951170111712183, 7.44053253339425069190783619267, 7.83317291651819864639250492514, 8.598341296363830787311762210788, 10.04861278183804889970414996523

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