Properties

Label 2-8016-1.1-c1-0-31
Degree $2$
Conductor $8016$
Sign $1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s + 9-s + 4·11-s + 2·13-s + 4·19-s + 4·21-s − 5·25-s − 27-s + 6·29-s − 8·31-s − 4·33-s + 6·37-s − 2·39-s − 2·43-s − 8·47-s + 9·49-s + 8·53-s − 4·57-s + 10·61-s − 4·63-s − 14·67-s + 12·71-s − 2·73-s + 5·75-s − 16·77-s − 2·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.917·19-s + 0.872·21-s − 25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.304·43-s − 1.16·47-s + 9/7·49-s + 1.09·53-s − 0.529·57-s + 1.28·61-s − 0.503·63-s − 1.71·67-s + 1.42·71-s − 0.234·73-s + 0.577·75-s − 1.82·77-s − 0.225·79-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: yes
Arithmetic: yes
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.309198561\)
\(L(\frac12)\) \(\approx\) \(1.309198561\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.68996557558252642783672355318, −6.89967187721694149878700621787, −6.45855797550976293851250943366, −5.90549662075292049802460164544, −5.19636634665212503382252581218, −4.06587320244688895933008338686, −3.65813846497706608760980555419, −2.82245760088355938139776327953, −1.58297242578836055475646597815, −0.60389804136970123600456008522, 0.60389804136970123600456008522, 1.58297242578836055475646597815, 2.82245760088355938139776327953, 3.65813846497706608760980555419, 4.06587320244688895933008338686, 5.19636634665212503382252581218, 5.90549662075292049802460164544, 6.45855797550976293851250943366, 6.89967187721694149878700621787, 7.68996557558252642783672355318

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