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 − 2.71·5-s + 0.775·7-s + 9-s − 3.70·11-s − 6.16·13-s + 2.71·15-s + 6.36·17-s − 0.922·19-s − 0.775·21-s + 4.47·23-s + 2.34·25-s − 27-s − 5.18·29-s − 1.35·31-s + 3.70·33-s − 2.10·35-s + 11.5·37-s + 6.16·39-s − 8.90·41-s − 5.52·43-s − 2.71·45-s − 8.07·47-s − 6.39·49-s − 6.36·51-s − 2.23·53-s + 10.0·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.21·5-s + 0.293·7-s + 0.333·9-s − 1.11·11-s − 1.71·13-s + 0.699·15-s + 1.54·17-s − 0.211·19-s − 0.169·21-s + 0.932·23-s + 0.469·25-s − 0.192·27-s − 0.962·29-s − 0.243·31-s + 0.645·33-s − 0.355·35-s + 1.89·37-s + 0.987·39-s − 1.39·41-s − 0.841·43-s − 0.404·45-s − 1.17·47-s − 0.914·49-s − 0.891·51-s − 0.307·53-s + 1.35·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4829814833$
$L(\frac12)$  $\approx$  $0.4829814833$
$L(\frac{3}{2})$   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 \( 1 + 2.71T + 5T^{2} \)
7 \( 1 - 0.775T + 7T^{2} \)
11 \( 1 + 3.70T + 11T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 + 0.922T + 19T^{2} \)
23 \( 1 - 4.47T + 23T^{2} \)
29 \( 1 + 5.18T + 29T^{2} \)
31 \( 1 + 1.35T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 8.90T + 41T^{2} \)
43 \( 1 + 5.52T + 43T^{2} \)
47 \( 1 + 8.07T + 47T^{2} \)
53 \( 1 + 2.23T + 53T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 + 5.08T + 61T^{2} \)
67 \( 1 + 0.628T + 67T^{2} \)
71 \( 1 + 3.07T + 71T^{2} \)
73 \( 1 - 3.83T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 - 4.76T + 83T^{2} \)
89 \( 1 - 0.390T + 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−7.62980836053410545173280156654, −7.50139134924448759435998044184, −6.52672404393075130932799115659, −5.55958336268288160368564618773, −4.93584451174440634173243538057, −4.56819532653169965771657561650, −3.43652765086006696861065627054, −2.86159104058633993753173393556, −1.66262567242642535821645263438, −0.34870196825135084370196334876, 0.34870196825135084370196334876, 1.66262567242642535821645263438, 2.86159104058633993753173393556, 3.43652765086006696861065627054, 4.56819532653169965771657561650, 4.93584451174440634173243538057, 5.55958336268288160368564618773, 6.52672404393075130932799115659, 7.50139134924448759435998044184, 7.62980836053410545173280156654

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