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 − 2.11·5-s + 0.802·7-s + 9-s + 1.53·11-s + 6.15·13-s − 2.11·15-s + 2.02·17-s + 8.09·19-s + 0.802·21-s + 1.69·23-s − 0.508·25-s + 27-s + 8.39·29-s − 1.83·31-s + 1.53·33-s − 1.70·35-s − 9.93·37-s + 6.15·39-s − 2.89·41-s − 2.94·43-s − 2.11·45-s + 7.81·47-s − 6.35·49-s + 2.02·51-s + 9.25·53-s − 3.24·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.947·5-s + 0.303·7-s + 0.333·9-s + 0.461·11-s + 1.70·13-s − 0.547·15-s + 0.491·17-s + 1.85·19-s + 0.175·21-s + 0.353·23-s − 0.101·25-s + 0.192·27-s + 1.55·29-s − 0.329·31-s + 0.266·33-s − 0.287·35-s − 1.63·37-s + 0.985·39-s − 0.451·41-s − 0.449·43-s − 0.315·45-s + 1.13·47-s − 0.907·49-s + 0.283·51-s + 1.27·53-s − 0.437·55-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  =  \((2,\ 8016,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.861704004\)
\(L(\frac12)\)  \(\approx\)  \(2.861704004\)
\(L(\frac{3}{2})\)   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 \( 1 + 2.11T + 5T^{2} \)
7 \( 1 - 0.802T + 7T^{2} \)
11 \( 1 - 1.53T + 11T^{2} \)
13 \( 1 - 6.15T + 13T^{2} \)
17 \( 1 - 2.02T + 17T^{2} \)
19 \( 1 - 8.09T + 19T^{2} \)
23 \( 1 - 1.69T + 23T^{2} \)
29 \( 1 - 8.39T + 29T^{2} \)
31 \( 1 + 1.83T + 31T^{2} \)
37 \( 1 + 9.93T + 37T^{2} \)
41 \( 1 + 2.89T + 41T^{2} \)
43 \( 1 + 2.94T + 43T^{2} \)
47 \( 1 - 7.81T + 47T^{2} \)
53 \( 1 - 9.25T + 53T^{2} \)
59 \( 1 + 5.76T + 59T^{2} \)
61 \( 1 + 0.0936T + 61T^{2} \)
67 \( 1 - 8.66T + 67T^{2} \)
71 \( 1 - 8.00T + 71T^{2} \)
73 \( 1 - 1.52T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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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

−7.918104781584161573201940445930, −7.23900220838192771116360732525, −6.64754311895873940000820258035, −5.69279843570697942907783611073, −4.99253673716103498019361730709, −4.02584938194663255283852766540, −3.53392494576885190478360987766, −2.96937809405993731816312546437, −1.58164215429320909551707958839, −0.902578889882153773795741859947, 0.902578889882153773795741859947, 1.58164215429320909551707958839, 2.96937809405993731816312546437, 3.53392494576885190478360987766, 4.02584938194663255283852766540, 4.99253673716103498019361730709, 5.69279843570697942907783611073, 6.64754311895873940000820258035, 7.23900220838192771116360732525, 7.918104781584161573201940445930

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