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 + 1.68·5-s − 0.722·7-s + 9-s − 4.92·11-s − 5.15·13-s − 1.68·15-s + 4.50·17-s − 3.23·19-s + 0.722·21-s − 9.11·23-s − 2.14·25-s − 27-s + 1.58·29-s + 2.13·31-s + 4.92·33-s − 1.21·35-s − 8.13·37-s + 5.15·39-s + 5.37·41-s + 4.74·43-s + 1.68·45-s + 3.74·47-s − 6.47·49-s − 4.50·51-s + 1.47·53-s − 8.31·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·5-s − 0.272·7-s + 0.333·9-s − 1.48·11-s − 1.43·13-s − 0.435·15-s + 1.09·17-s − 0.741·19-s + 0.157·21-s − 1.90·23-s − 0.429·25-s − 0.192·27-s + 0.293·29-s + 0.383·31-s + 0.857·33-s − 0.206·35-s − 1.33·37-s + 0.825·39-s + 0.839·41-s + 0.723·43-s + 0.251·45-s + 0.546·47-s − 0.925·49-s − 0.630·51-s + 0.202·53-s − 1.12·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9286599749$
$L(\frac12)$  $\approx$  $0.9286599749$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 - 1.68T + 5T^{2} \)
7 \( 1 + 0.722T + 7T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
13 \( 1 + 5.15T + 13T^{2} \)
17 \( 1 - 4.50T + 17T^{2} \)
19 \( 1 + 3.23T + 19T^{2} \)
23 \( 1 + 9.11T + 23T^{2} \)
29 \( 1 - 1.58T + 29T^{2} \)
31 \( 1 - 2.13T + 31T^{2} \)
37 \( 1 + 8.13T + 37T^{2} \)
41 \( 1 - 5.37T + 41T^{2} \)
43 \( 1 - 4.74T + 43T^{2} \)
47 \( 1 - 3.74T + 47T^{2} \)
53 \( 1 - 1.47T + 53T^{2} \)
59 \( 1 + 3.12T + 59T^{2} \)
61 \( 1 - 9.12T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + 2.51T + 73T^{2} \)
79 \( 1 - 12.4T + 79T^{2} \)
83 \( 1 - 2.96T + 83T^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−7.77131251551152383827071859036, −7.19998001919254685737594910527, −6.23569789031746598853315762511, −5.76664121632894636375667781244, −5.16285548015989648338853417034, −4.50621473444740034649585240633, −3.47980234319214260784952624541, −2.43106412704242453815552797767, −1.98420257226293536166453664776, −0.46108285108728015078980882500, 0.46108285108728015078980882500, 1.98420257226293536166453664776, 2.43106412704242453815552797767, 3.47980234319214260784952624541, 4.50621473444740034649585240633, 5.16285548015989648338853417034, 5.76664121632894636375667781244, 6.23569789031746598853315762511, 7.19998001919254685737594910527, 7.77131251551152383827071859036

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