Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
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 + 0.957·5-s + 0.898·7-s + 9-s + 0.936·11-s + 1.86·13-s + 0.957·15-s + 6.88·17-s − 0.644·19-s + 0.898·21-s + 1.95·23-s − 4.08·25-s + 27-s − 0.602·29-s − 3.41·31-s + 0.936·33-s + 0.860·35-s + 11.6·37-s + 1.86·39-s + 0.378·41-s + 8.52·43-s + 0.957·45-s + 1.26·47-s − 6.19·49-s + 6.88·51-s − 9.54·53-s + 0.896·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.428·5-s + 0.339·7-s + 0.333·9-s + 0.282·11-s + 0.517·13-s + 0.247·15-s + 1.67·17-s − 0.147·19-s + 0.196·21-s + 0.408·23-s − 0.816·25-s + 0.192·27-s − 0.111·29-s − 0.613·31-s + 0.163·33-s + 0.145·35-s + 1.92·37-s + 0.298·39-s + 0.0591·41-s + 1.30·43-s + 0.142·45-s + 0.184·47-s − 0.884·49-s + 0.964·51-s − 1.31·53-s + 0.120·55-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  =  \(0\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.476309068\)
\(L(\frac12)\)  \(\approx\)  \(3.476309068\)
\(L(\frac{3}{2})\)   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 \( 1 - 0.957T + 5T^{2} \)
7 \( 1 - 0.898T + 7T^{2} \)
11 \( 1 - 0.936T + 11T^{2} \)
13 \( 1 - 1.86T + 13T^{2} \)
17 \( 1 - 6.88T + 17T^{2} \)
19 \( 1 + 0.644T + 19T^{2} \)
23 \( 1 - 1.95T + 23T^{2} \)
29 \( 1 + 0.602T + 29T^{2} \)
31 \( 1 + 3.41T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 - 0.378T + 41T^{2} \)
43 \( 1 - 8.52T + 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + 9.54T + 53T^{2} \)
59 \( 1 - 1.15T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 7.35T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 0.169T + 73T^{2} \)
79 \( 1 - 3.88T + 79T^{2} \)
83 \( 1 - 3.18T + 83T^{2} \)
89 \( 1 - 6.13T + 89T^{2} \)
97 \( 1 - 6.69T + 97T^{2} \)
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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

−7.73573943127987451264580293096, −7.44321795147573113742343234440, −6.30084213527565259739508869584, −5.86058620761414541608431768488, −5.05164570434845442949385373132, −4.17007284528968430880905282454, −3.48961658843944869987641813901, −2.69230019989115336412967520052, −1.75010491873923793225083766154, −0.964009810244240271423443631255, 0.964009810244240271423443631255, 1.75010491873923793225083766154, 2.69230019989115336412967520052, 3.48961658843944869987641813901, 4.17007284528968430880905282454, 5.05164570434845442949385373132, 5.86058620761414541608431768488, 6.30084213527565259739508869584, 7.44321795147573113742343234440, 7.73573943127987451264580293096

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