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 − 4.17·5-s + 1.78·7-s + 9-s − 4.35·11-s + 2.54·13-s + 4.17·15-s + 0.0716·17-s + 7.16·19-s − 1.78·21-s + 2.75·23-s + 12.3·25-s − 27-s + 5.44·29-s − 3.34·31-s + 4.35·33-s − 7.45·35-s − 6.04·37-s − 2.54·39-s + 3.33·41-s − 3.47·43-s − 4.17·45-s − 1.84·47-s − 3.80·49-s − 0.0716·51-s − 10.4·53-s + 18.1·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.86·5-s + 0.675·7-s + 0.333·9-s − 1.31·11-s + 0.707·13-s + 1.07·15-s + 0.0173·17-s + 1.64·19-s − 0.389·21-s + 0.573·23-s + 2.47·25-s − 0.192·27-s + 1.01·29-s − 0.600·31-s + 0.757·33-s − 1.25·35-s − 0.993·37-s − 0.408·39-s + 0.520·41-s − 0.530·43-s − 0.621·45-s − 0.269·47-s − 0.544·49-s − 0.0100·51-s − 1.43·53-s + 2.44·55-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

\(L(1)\) \(\approx\) \(0.8889289178\)
\(L(\frac12)\) \(\approx\) \(0.8889289178\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.17T + 5T^{2} \)
7 \( 1 - 1.78T + 7T^{2} \)
11 \( 1 + 4.35T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 - 0.0716T + 17T^{2} \)
19 \( 1 - 7.16T + 19T^{2} \)
23 \( 1 - 2.75T + 23T^{2} \)
29 \( 1 - 5.44T + 29T^{2} \)
31 \( 1 + 3.34T + 31T^{2} \)
37 \( 1 + 6.04T + 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 + 3.47T + 43T^{2} \)
47 \( 1 + 1.84T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 7.84T + 59T^{2} \)
61 \( 1 + 6.59T + 61T^{2} \)
67 \( 1 + 3.06T + 67T^{2} \)
71 \( 1 - 9.41T + 71T^{2} \)
73 \( 1 + 8.85T + 73T^{2} \)
79 \( 1 + 0.0583T + 79T^{2} \)
83 \( 1 + 0.584T + 83T^{2} \)
89 \( 1 + 3.61T + 89T^{2} \)
97 \( 1 - 0.334T + 97T^{2} \)
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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

−7.72942292198783085930495788804, −7.39770962169117931886768318406, −6.58859280976996216833186292775, −5.56030466284292115970953767528, −4.89033606854895219328681643416, −4.51415597183960748120235249334, −3.40521420608671988332538580080, −3.04409538108740098117941355544, −1.48599722487777907647564550268, −0.50927216099690422977278035870, 0.50927216099690422977278035870, 1.48599722487777907647564550268, 3.04409538108740098117941355544, 3.40521420608671988332538580080, 4.51415597183960748120235249334, 4.89033606854895219328681643416, 5.56030466284292115970953767528, 6.58859280976996216833186292775, 7.39770962169117931886768318406, 7.72942292198783085930495788804

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