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 + 2·5-s + 4·7-s + 9-s + 4·11-s − 2·15-s − 4·17-s + 4·19-s − 4·21-s + 4·23-s − 25-s − 27-s − 2·29-s − 4·31-s − 4·33-s + 8·35-s − 12·37-s + 12·41-s + 8·43-s + 2·45-s + 9·49-s + 4·51-s + 14·53-s + 8·55-s − 4·57-s − 2·59-s − 2·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.516·15-s − 0.970·17-s + 0.917·19-s − 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s + 1.35·35-s − 1.97·37-s + 1.87·41-s + 1.21·43-s + 0.298·45-s + 9/7·49-s + 0.560·51-s + 1.92·53-s + 1.07·55-s − 0.529·57-s − 0.260·59-s − 0.256·61-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  :  $\chi_{8016} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.882635359$
$L(\frac12)$  $\approx$  $2.882635359$
$L(\frac{3}{2})$   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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−17.22114254031134, −16.60580358741192, −15.80046702935358, −15.21632912027219, −14.46919827092772, −14.14130029936468, −13.57868076304410, −12.86888569632734, −12.07055522486982, −11.63931831271572, −10.97051493776288, −10.67421683588995, −9.709417225937053, −9.059464703084359, −8.747691606707752, −7.623027758762913, −7.167218532225350, −6.389416302013473, −5.621402424665108, −5.192427812073934, −4.399798581762016, −3.721894945586460, −2.361953204921729, −1.681542441069289, −0.9466354039962679, 0.9466354039962679, 1.681542441069289, 2.361953204921729, 3.721894945586460, 4.399798581762016, 5.192427812073934, 5.621402424665108, 6.389416302013473, 7.167218532225350, 7.623027758762913, 8.747691606707752, 9.059464703084359, 9.709417225937053, 10.67421683588995, 10.97051493776288, 11.63931831271572, 12.07055522486982, 12.86888569632734, 13.57868076304410, 14.14130029936468, 14.46919827092772, 15.21632912027219, 15.80046702935358, 16.60580358741192, 17.22114254031134

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