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 − 2.04·5-s − 1.81·7-s + 9-s + 1.62·11-s − 3.43·13-s + 2.04·15-s − 4.77·17-s − 5.04·19-s + 1.81·21-s − 3.71·23-s − 0.817·25-s − 27-s + 5.69·29-s − 4.23·31-s − 1.62·33-s + 3.70·35-s − 2.11·37-s + 3.43·39-s + 2.68·41-s − 12.6·43-s − 2.04·45-s + 8.44·47-s − 3.71·49-s + 4.77·51-s − 2.62·53-s − 3.31·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.914·5-s − 0.685·7-s + 0.333·9-s + 0.488·11-s − 0.952·13-s + 0.528·15-s − 1.15·17-s − 1.15·19-s + 0.395·21-s − 0.775·23-s − 0.163·25-s − 0.192·27-s + 1.05·29-s − 0.760·31-s − 0.282·33-s + 0.626·35-s − 0.347·37-s + 0.550·39-s + 0.419·41-s − 1.93·43-s − 0.304·45-s + 1.23·47-s − 0.530·49-s + 0.668·51-s − 0.360·53-s − 0.447·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2293209862$
$L(\frac12)$  $\approx$  $0.2293209862$
$L(\frac{3}{2})$   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 \( 1 + 2.04T + 5T^{2} \)
7 \( 1 + 1.81T + 7T^{2} \)
11 \( 1 - 1.62T + 11T^{2} \)
13 \( 1 + 3.43T + 13T^{2} \)
17 \( 1 + 4.77T + 17T^{2} \)
19 \( 1 + 5.04T + 19T^{2} \)
23 \( 1 + 3.71T + 23T^{2} \)
29 \( 1 - 5.69T + 29T^{2} \)
31 \( 1 + 4.23T + 31T^{2} \)
37 \( 1 + 2.11T + 37T^{2} \)
41 \( 1 - 2.68T + 41T^{2} \)
43 \( 1 + 12.6T + 43T^{2} \)
47 \( 1 - 8.44T + 47T^{2} \)
53 \( 1 + 2.62T + 53T^{2} \)
59 \( 1 + 0.400T + 59T^{2} \)
61 \( 1 - 0.131T + 61T^{2} \)
67 \( 1 + 6.63T + 67T^{2} \)
71 \( 1 + 12.2T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 2.93T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 - 5.43T + 89T^{2} \)
97 \( 1 - 9.62T + 97T^{2} \)
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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

−7.76306856105695208961984670153, −6.98165249905407968252436149042, −6.56547956520159491637689012598, −5.89050943120343850163119853653, −4.87366115282883470618508977774, −4.29725518435757785481698152602, −3.71084750407986371310474300711, −2.69236826525571230886322117244, −1.75212507987232970629288852219, −0.23363783267227431755367386218, 0.23363783267227431755367386218, 1.75212507987232970629288852219, 2.69236826525571230886322117244, 3.71084750407986371310474300711, 4.29725518435757785481698152602, 4.87366115282883470618508977774, 5.89050943120343850163119853653, 6.56547956520159491637689012598, 6.98165249905407968252436149042, 7.76306856105695208961984670153

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