Properties

Degree $2$
Conductor $8016$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3.01·5-s + 1.84·7-s + 9-s + 0.0702·11-s + 1.68·13-s − 3.01·15-s − 2.17·17-s + 2.90·19-s + 1.84·21-s − 1.67·23-s + 4.09·25-s + 27-s − 4.24·29-s + 4.28·31-s + 0.0702·33-s − 5.56·35-s − 10.8·37-s + 1.68·39-s − 7.20·41-s + 4.33·43-s − 3.01·45-s − 11.5·47-s − 3.58·49-s − 2.17·51-s + 3.88·53-s − 0.211·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s + 0.698·7-s + 0.333·9-s + 0.0211·11-s + 0.468·13-s − 0.778·15-s − 0.527·17-s + 0.667·19-s + 0.402·21-s − 0.348·23-s + 0.818·25-s + 0.192·27-s − 0.787·29-s + 0.770·31-s + 0.0122·33-s − 0.941·35-s − 1.78·37-s + 0.270·39-s − 1.12·41-s + 0.661·43-s − 0.449·45-s − 1.69·47-s − 0.512·49-s − 0.304·51-s + 0.533·53-s − 0.0285·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 + 3.01T + 5T^{2} \)
7 \( 1 - 1.84T + 7T^{2} \)
11 \( 1 - 0.0702T + 11T^{2} \)
13 \( 1 - 1.68T + 13T^{2} \)
17 \( 1 + 2.17T + 17T^{2} \)
19 \( 1 - 2.90T + 19T^{2} \)
23 \( 1 + 1.67T + 23T^{2} \)
29 \( 1 + 4.24T + 29T^{2} \)
31 \( 1 - 4.28T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 7.20T + 41T^{2} \)
43 \( 1 - 4.33T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 3.88T + 53T^{2} \)
59 \( 1 + 5.40T + 59T^{2} \)
61 \( 1 - 2.01T + 61T^{2} \)
67 \( 1 - 6.98T + 67T^{2} \)
71 \( 1 + 2.15T + 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 9.35T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 - 13.2T + 97T^{2} \)
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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

−7.65214341231707553362597166622, −7.00714751877792957553948221544, −6.25359863513611130901703655974, −5.10881161534689860738369355411, −4.65167435266158513429183501276, −3.65755248688920706377624264153, −3.44081000747088007639815164738, −2.22205001649635053201401665423, −1.31103109483073760969153543524, 0, 1.31103109483073760969153543524, 2.22205001649635053201401665423, 3.44081000747088007639815164738, 3.65755248688920706377624264153, 4.65167435266158513429183501276, 5.10881161534689860738369355411, 6.25359863513611130901703655974, 7.00714751877792957553948221544, 7.65214341231707553362597166622

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