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 + 3.34·5-s − 3.54·7-s + 9-s + 5.22·11-s + 5.27·13-s + 3.34·15-s + 6.86·17-s − 5.25·19-s − 3.54·21-s − 2.32·23-s + 6.21·25-s + 27-s − 0.922·29-s + 10.9·31-s + 5.22·33-s − 11.8·35-s − 10.2·37-s + 5.27·39-s + 2.58·41-s − 4.38·43-s + 3.34·45-s − 6.30·47-s + 5.57·49-s + 6.86·51-s + 10.7·53-s + 17.4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.49·5-s − 1.34·7-s + 0.333·9-s + 1.57·11-s + 1.46·13-s + 0.864·15-s + 1.66·17-s − 1.20·19-s − 0.773·21-s − 0.485·23-s + 1.24·25-s + 0.192·27-s − 0.171·29-s + 1.97·31-s + 0.908·33-s − 2.00·35-s − 1.67·37-s + 0.844·39-s + 0.403·41-s − 0.668·43-s + 0.499·45-s − 0.919·47-s + 0.796·49-s + 0.961·51-s + 1.47·53-s + 2.35·55-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

\(L(1)\) \(\approx\) \(3.900799942\)
\(L(\frac12)\) \(\approx\) \(3.900799942\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.34T + 5T^{2} \)
7 \( 1 + 3.54T + 7T^{2} \)
11 \( 1 - 5.22T + 11T^{2} \)
13 \( 1 - 5.27T + 13T^{2} \)
17 \( 1 - 6.86T + 17T^{2} \)
19 \( 1 + 5.25T + 19T^{2} \)
23 \( 1 + 2.32T + 23T^{2} \)
29 \( 1 + 0.922T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 + 4.38T + 43T^{2} \)
47 \( 1 + 6.30T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 8.19T + 67T^{2} \)
71 \( 1 - 3.27T + 71T^{2} \)
73 \( 1 + 9.70T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 1.83T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−8.012794548343608818882969716392, −6.69340369129640557197726077167, −6.49864889592226765037721079653, −6.03497449898355508047010917089, −5.20520099322514756880623591875, −3.91349660960156895361155467313, −3.57431307100996649193135616538, −2.70176686710468921109303483584, −1.71913000739652613781359198547, −1.04359694346516040990901997101, 1.04359694346516040990901997101, 1.71913000739652613781359198547, 2.70176686710468921109303483584, 3.57431307100996649193135616538, 3.91349660960156895361155467313, 5.20520099322514756880623591875, 6.03497449898355508047010917089, 6.49864889592226765037721079653, 6.69340369129640557197726077167, 8.012794548343608818882969716392

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