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 + 3.38·5-s − 2.18·7-s + 9-s − 1.42·11-s + 5.00·13-s − 3.38·15-s + 5.50·17-s + 7.82·19-s + 2.18·21-s − 7.75·23-s + 6.43·25-s − 27-s + 6.29·29-s + 9.08·31-s + 1.42·33-s − 7.37·35-s + 4.72·37-s − 5.00·39-s − 11.8·41-s + 7.11·43-s + 3.38·45-s + 8.21·47-s − 2.24·49-s − 5.50·51-s − 10.8·53-s − 4.81·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·5-s − 0.824·7-s + 0.333·9-s − 0.429·11-s + 1.38·13-s − 0.873·15-s + 1.33·17-s + 1.79·19-s + 0.475·21-s − 1.61·23-s + 1.28·25-s − 0.192·27-s + 1.16·29-s + 1.63·31-s + 0.247·33-s − 1.24·35-s + 0.776·37-s − 0.800·39-s − 1.84·41-s + 1.08·43-s + 0.504·45-s + 1.19·47-s − 0.321·49-s − 0.771·51-s − 1.48·53-s − 0.648·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.530508964$
$L(\frac12)$  $\approx$  $2.530508964$
$L(\frac{3}{2})$   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 \( 1 - 3.38T + 5T^{2} \)
7 \( 1 + 2.18T + 7T^{2} \)
11 \( 1 + 1.42T + 11T^{2} \)
13 \( 1 - 5.00T + 13T^{2} \)
17 \( 1 - 5.50T + 17T^{2} \)
19 \( 1 - 7.82T + 19T^{2} \)
23 \( 1 + 7.75T + 23T^{2} \)
29 \( 1 - 6.29T + 29T^{2} \)
31 \( 1 - 9.08T + 31T^{2} \)
37 \( 1 - 4.72T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 7.11T + 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 6.83T + 59T^{2} \)
61 \( 1 + 1.73T + 61T^{2} \)
67 \( 1 - 1.61T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 - 4.43T + 73T^{2} \)
79 \( 1 + 4.26T + 79T^{2} \)
83 \( 1 + 4.41T + 83T^{2} \)
89 \( 1 + 8.56T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−7.86283717527507095473858361643, −6.90331657167126634962367253465, −6.19403262233540269815403690217, −5.84953669539834953516329843187, −5.37381639244143410119251053554, −4.39058092121893405136286216033, −3.32184565185433132860388510853, −2.77464963390590060339881430088, −1.55543027547813372653103583052, −0.888596868404818975717313597860, 0.888596868404818975717313597860, 1.55543027547813372653103583052, 2.77464963390590060339881430088, 3.32184565185433132860388510853, 4.39058092121893405136286216033, 5.37381639244143410119251053554, 5.84953669539834953516329843187, 6.19403262233540269815403690217, 6.90331657167126634962367253465, 7.86283717527507095473858361643

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