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 − 0.597·5-s − 2.60·7-s + 9-s + 4.70·11-s − 4.30·13-s − 0.597·15-s + 2.94·17-s + 4.08·19-s − 2.60·21-s − 2.33·23-s − 4.64·25-s + 27-s − 2.00·29-s − 5.83·31-s + 4.70·33-s + 1.55·35-s − 1.82·37-s − 4.30·39-s + 8.73·41-s − 11.8·43-s − 0.597·45-s + 2.91·47-s − 0.188·49-s + 2.94·51-s + 9.63·53-s − 2.80·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.267·5-s − 0.986·7-s + 0.333·9-s + 1.41·11-s − 1.19·13-s − 0.154·15-s + 0.713·17-s + 0.937·19-s − 0.569·21-s − 0.486·23-s − 0.928·25-s + 0.192·27-s − 0.372·29-s − 1.04·31-s + 0.818·33-s + 0.263·35-s − 0.299·37-s − 0.689·39-s + 1.36·41-s − 1.80·43-s − 0.0890·45-s + 0.425·47-s − 0.0268·49-s + 0.411·51-s + 1.32·53-s − 0.378·55-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 \( 1 + 0.597T + 5T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 - 4.70T + 11T^{2} \)
13 \( 1 + 4.30T + 13T^{2} \)
17 \( 1 - 2.94T + 17T^{2} \)
19 \( 1 - 4.08T + 19T^{2} \)
23 \( 1 + 2.33T + 23T^{2} \)
29 \( 1 + 2.00T + 29T^{2} \)
31 \( 1 + 5.83T + 31T^{2} \)
37 \( 1 + 1.82T + 37T^{2} \)
41 \( 1 - 8.73T + 41T^{2} \)
43 \( 1 + 11.8T + 43T^{2} \)
47 \( 1 - 2.91T + 47T^{2} \)
53 \( 1 - 9.63T + 53T^{2} \)
59 \( 1 + 6.28T + 59T^{2} \)
61 \( 1 + 8.58T + 61T^{2} \)
67 \( 1 - 5.46T + 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 + 2.56T + 73T^{2} \)
79 \( 1 + 4.32T + 79T^{2} \)
83 \( 1 - 8.40T + 83T^{2} \)
89 \( 1 - 8.59T + 89T^{2} \)
97 \( 1 - 6.10T + 97T^{2} \)
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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

−7.43987477364225557985027284353, −6.97616555139208995602764677430, −6.15681777093388384731127364147, −5.47435887602262805877620869054, −4.49195556572445110529444672284, −3.68101255315542346444958968872, −3.30494778613244330455351009633, −2.29704730953931081890847018502, −1.32844109365442934920119690437, 0, 1.32844109365442934920119690437, 2.29704730953931081890847018502, 3.30494778613244330455351009633, 3.68101255315542346444958968872, 4.49195556572445110529444672284, 5.47435887602262805877620869054, 6.15681777093388384731127364147, 6.97616555139208995602764677430, 7.43987477364225557985027284353

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