Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 3.61·5-s − 2.72·7-s + 9-s − 4.28·11-s − 0.0474·13-s + 3.61·15-s + 3.70·17-s − 0.502·19-s − 2.72·21-s − 2.97·23-s + 8.08·25-s + 27-s − 6.25·29-s − 5.93·31-s − 4.28·33-s − 9.86·35-s + 0.158·37-s − 0.0474·39-s − 3.89·41-s − 4.94·43-s + 3.61·45-s − 8.39·47-s + 0.440·49-s + 3.70·51-s − 10.7·53-s − 15.4·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.61·5-s − 1.03·7-s + 0.333·9-s − 1.29·11-s − 0.0131·13-s + 0.933·15-s + 0.899·17-s − 0.115·19-s − 0.595·21-s − 0.621·23-s + 1.61·25-s + 0.192·27-s − 1.16·29-s − 1.06·31-s − 0.745·33-s − 1.66·35-s + 0.0260·37-s − 0.00759·39-s − 0.608·41-s − 0.754·43-s + 0.539·45-s − 1.22·47-s + 0.0629·49-s + 0.519·51-s − 1.47·53-s − 2.08·55-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  =  \(1\)
Selberg data  =  \((2,\ 8016,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 - 3.61T + 5T^{2} \)
7 \( 1 + 2.72T + 7T^{2} \)
11 \( 1 + 4.28T + 11T^{2} \)
13 \( 1 + 0.0474T + 13T^{2} \)
17 \( 1 - 3.70T + 17T^{2} \)
19 \( 1 + 0.502T + 19T^{2} \)
23 \( 1 + 2.97T + 23T^{2} \)
29 \( 1 + 6.25T + 29T^{2} \)
31 \( 1 + 5.93T + 31T^{2} \)
37 \( 1 - 0.158T + 37T^{2} \)
41 \( 1 + 3.89T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 + 8.39T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 3.54T + 59T^{2} \)
61 \( 1 - 3.62T + 61T^{2} \)
67 \( 1 + 0.477T + 67T^{2} \)
71 \( 1 + 0.963T + 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 - 1.53T + 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 - 1.01T + 89T^{2} \)
97 \( 1 + 1.19T + 97T^{2} \)
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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

−7.52968839904298485990094843645, −6.71329684970582607989592406928, −6.05992753285001037956459271247, −5.48576941860191337747791706975, −4.88823934535730197050297189554, −3.58037185492773101395328845762, −3.06234103453882132118329169449, −2.22164552089705022849597504044, −1.58933089237121421702711929855, 0, 1.58933089237121421702711929855, 2.22164552089705022849597504044, 3.06234103453882132118329169449, 3.58037185492773101395328845762, 4.88823934535730197050297189554, 5.48576941860191337747791706975, 6.05992753285001037956459271247, 6.71329684970582607989592406928, 7.52968839904298485990094843645

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