Properties

Degree $2$
Conductor $8016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.413·5-s − 2.72·7-s + 9-s + 3.29·11-s − 3.51·13-s − 0.413·15-s − 2.04·17-s − 0.104·19-s + 2.72·21-s + 7.14·23-s − 4.82·25-s − 27-s + 3.94·29-s + 6.72·31-s − 3.29·33-s − 1.12·35-s − 11.8·37-s + 3.51·39-s + 1.68·41-s + 3.25·43-s + 0.413·45-s + 1.28·47-s + 0.427·49-s + 2.04·51-s − 2.39·53-s + 1.36·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.185·5-s − 1.03·7-s + 0.333·9-s + 0.992·11-s − 0.974·13-s − 0.106·15-s − 0.495·17-s − 0.0239·19-s + 0.594·21-s + 1.49·23-s − 0.965·25-s − 0.192·27-s + 0.733·29-s + 1.20·31-s − 0.573·33-s − 0.190·35-s − 1.94·37-s + 0.562·39-s + 0.263·41-s + 0.495·43-s + 0.0616·45-s + 0.187·47-s + 0.0610·49-s + 0.286·51-s − 0.328·53-s + 0.183·55-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: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183897914\)
\(L(\frac12)\) \(\approx\) \(1.183897914\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.413T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 - 3.29T + 11T^{2} \)
13 \( 1 + 3.51T + 13T^{2} \)
17 \( 1 + 2.04T + 17T^{2} \)
19 \( 1 + 0.104T + 19T^{2} \)
23 \( 1 - 7.14T + 23T^{2} \)
29 \( 1 - 3.94T + 29T^{2} \)
31 \( 1 - 6.72T + 31T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 - 3.25T + 43T^{2} \)
47 \( 1 - 1.28T + 47T^{2} \)
53 \( 1 + 2.39T + 53T^{2} \)
59 \( 1 + 4.83T + 59T^{2} \)
61 \( 1 + 0.869T + 61T^{2} \)
67 \( 1 + 0.884T + 67T^{2} \)
71 \( 1 + 5.76T + 71T^{2} \)
73 \( 1 + 9.55T + 73T^{2} \)
79 \( 1 + 8.30T + 79T^{2} \)
83 \( 1 - 9.73T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 15.3T + 97T^{2} \)
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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

−7.59587743631787700905612679156, −6.97717134079831278223617495222, −6.46957223782778154542139691337, −5.91918431518564819045339850216, −4.97643093237293274485208355411, −4.43039431563634193697036097571, −3.47271614792183557963162627087, −2.75438836864797443148529596704, −1.68243162750770505510340117892, −0.55464746413340524569983828518, 0.55464746413340524569983828518, 1.68243162750770505510340117892, 2.75438836864797443148529596704, 3.47271614792183557963162627087, 4.43039431563634193697036097571, 4.97643093237293274485208355411, 5.91918431518564819045339850216, 6.46957223782778154542139691337, 6.97717134079831278223617495222, 7.59587743631787700905612679156

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