Properties

Label 2-8016-1.1-c1-0-70
Degree $2$
Conductor $8016$
Sign $1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.87·5-s + 3.28·7-s + 9-s + 4.43·13-s − 1.87·15-s + 6.72·17-s − 1.69·19-s + 3.28·21-s + 1.69·23-s − 1.47·25-s + 27-s + 9.75·29-s − 1.59·31-s − 6.16·35-s + 4.53·37-s + 4.43·39-s + 3.33·41-s − 7.16·43-s − 1.87·45-s − 4.47·47-s + 3.77·49-s + 6.72·51-s − 0.184·53-s − 1.69·57-s + 13.4·59-s − 5.75·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.839·5-s + 1.24·7-s + 0.333·9-s + 1.23·13-s − 0.484·15-s + 1.63·17-s − 0.388·19-s + 0.716·21-s + 0.353·23-s − 0.294·25-s + 0.192·27-s + 1.81·29-s − 0.285·31-s − 1.04·35-s + 0.746·37-s + 0.710·39-s + 0.521·41-s − 1.09·43-s − 0.279·45-s − 0.652·47-s + 0.539·49-s + 0.941·51-s − 0.0253·53-s − 0.224·57-s + 1.75·59-s − 0.736·61-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: $1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.114244850\)
\(L(\frac12)\) \(\approx\) \(3.114244850\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.87T + 5T^{2} \)
7 \( 1 - 3.28T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4.43T + 13T^{2} \)
17 \( 1 - 6.72T + 17T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 - 1.69T + 23T^{2} \)
29 \( 1 - 9.75T + 29T^{2} \)
31 \( 1 + 1.59T + 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 + 7.16T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 + 0.184T + 53T^{2} \)
59 \( 1 - 13.4T + 59T^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
67 \( 1 - 8.45T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−8.129264334531284823372041251469, −7.38532530552715210916963883516, −6.54266192841143731547907270407, −5.69553977292108452936618179775, −4.89660755379957055463950842754, −4.20463452505005421532302586334, −3.55492863029438337318198097578, −2.80804388058395370196172593860, −1.63085657376649837564411545998, −0.937569289000548086375443105324, 0.937569289000548086375443105324, 1.63085657376649837564411545998, 2.80804388058395370196172593860, 3.55492863029438337318198097578, 4.20463452505005421532302586334, 4.89660755379957055463950842754, 5.69553977292108452936618179775, 6.54266192841143731547907270407, 7.38532530552715210916963883516, 8.129264334531284823372041251469

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