Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
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 − 3.20·5-s − 3.68·7-s + 9-s − 2.41·11-s − 0.292·13-s − 3.20·15-s + 7.19·17-s + 2.96·19-s − 3.68·21-s − 5.50·23-s + 5.28·25-s + 27-s + 3.32·29-s + 3.32·31-s − 2.41·33-s + 11.8·35-s − 0.357·37-s − 0.292·39-s + 3.49·41-s + 11.1·43-s − 3.20·45-s + 0.406·47-s + 6.59·49-s + 7.19·51-s − 10.6·53-s + 7.73·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.43·5-s − 1.39·7-s + 0.333·9-s − 0.727·11-s − 0.0810·13-s − 0.828·15-s + 1.74·17-s + 0.680·19-s − 0.804·21-s − 1.14·23-s + 1.05·25-s + 0.192·27-s + 0.616·29-s + 0.597·31-s − 0.419·33-s + 1.99·35-s − 0.0587·37-s − 0.0468·39-s + 0.546·41-s + 1.69·43-s − 0.478·45-s + 0.0593·47-s + 0.942·49-s + 1.00·51-s − 1.46·53-s + 1.04·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;167\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
167 \( 1 - T \)
good5 \( 1 + 3.20T + 5T^{2} \)
7 \( 1 + 3.68T + 7T^{2} \)
11 \( 1 + 2.41T + 11T^{2} \)
13 \( 1 + 0.292T + 13T^{2} \)
17 \( 1 - 7.19T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
23 \( 1 + 5.50T + 23T^{2} \)
29 \( 1 - 3.32T + 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 + 0.357T + 37T^{2} \)
41 \( 1 - 3.49T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 0.406T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 - 9.48T + 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 + 2.68T + 71T^{2} \)
73 \( 1 + 6.27T + 73T^{2} \)
79 \( 1 + 6.66T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + 4.51T + 89T^{2} \)
97 \( 1 - 1.77T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.70909221444122392259669913202, −7.02885016568589479083637620048, −6.13269021928201127199119163210, −5.44941518726886528733752112378, −4.39443460744983577897792250520, −3.76447512690177841282856997774, −3.13841879255035874565313779878, −2.64053315201547211358656128563, −1.04699263015907031249581359895, 0, 1.04699263015907031249581359895, 2.64053315201547211358656128563, 3.13841879255035874565313779878, 3.76447512690177841282856997774, 4.39443460744983577897792250520, 5.44941518726886528733752112378, 6.13269021928201127199119163210, 7.02885016568589479083637620048, 7.70909221444122392259669913202

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