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 − 1.85·5-s − 2.41·7-s + 9-s + 2.49·11-s + 0.116·13-s + 1.85·15-s − 6.12·17-s + 3.34·19-s + 2.41·21-s + 8.19·23-s − 1.57·25-s − 27-s + 0.0936·29-s − 1.44·31-s − 2.49·33-s + 4.47·35-s − 0.230·37-s − 0.116·39-s − 4.49·41-s − 5.35·43-s − 1.85·45-s − 7.01·47-s − 1.16·49-s + 6.12·51-s + 3.67·53-s − 4.62·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.828·5-s − 0.913·7-s + 0.333·9-s + 0.752·11-s + 0.0322·13-s + 0.478·15-s − 1.48·17-s + 0.766·19-s + 0.527·21-s + 1.70·23-s − 0.314·25-s − 0.192·27-s + 0.0173·29-s − 0.260·31-s − 0.434·33-s + 0.756·35-s − 0.0378·37-s − 0.0186·39-s − 0.702·41-s − 0.816·43-s − 0.276·45-s − 1.02·47-s − 0.166·49-s + 0.858·51-s + 0.505·53-s − 0.623·55-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 \( 1 + 1.85T + 5T^{2} \)
7 \( 1 + 2.41T + 7T^{2} \)
11 \( 1 - 2.49T + 11T^{2} \)
13 \( 1 - 0.116T + 13T^{2} \)
17 \( 1 + 6.12T + 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 - 8.19T + 23T^{2} \)
29 \( 1 - 0.0936T + 29T^{2} \)
31 \( 1 + 1.44T + 31T^{2} \)
37 \( 1 + 0.230T + 37T^{2} \)
41 \( 1 + 4.49T + 41T^{2} \)
43 \( 1 + 5.35T + 43T^{2} \)
47 \( 1 + 7.01T + 47T^{2} \)
53 \( 1 - 3.67T + 53T^{2} \)
59 \( 1 - 1.76T + 59T^{2} \)
61 \( 1 - 8.66T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 - 8.51T + 71T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 - 3.24T + 79T^{2} \)
83 \( 1 - 0.437T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 5.72T + 97T^{2} \)
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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

−7.24697543291405337555318986250, −6.73017499953512564032654218011, −6.38628890733383855755219368142, −5.27121414775428072436993408467, −4.74759158009445305793912303233, −3.76168615374351324035760889045, −3.39300883804781930132267786355, −2.23599146688793544886283487251, −0.995191673180787130142209744791, 0, 0.995191673180787130142209744791, 2.23599146688793544886283487251, 3.39300883804781930132267786355, 3.76168615374351324035760889045, 4.74759158009445305793912303233, 5.27121414775428072436993408467, 6.38628890733383855755219368142, 6.73017499953512564032654218011, 7.24697543291405337555318986250

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