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 + 4.20·5-s + 0.497·7-s + 9-s + 0.310·11-s − 0.223·13-s − 4.20·15-s + 3.33·17-s + 0.261·19-s − 0.497·21-s + 9.01·23-s + 12.6·25-s − 27-s + 3.87·29-s − 0.329·31-s − 0.310·33-s + 2.09·35-s − 0.490·37-s + 0.223·39-s + 9.61·41-s − 2.34·43-s + 4.20·45-s − 9.94·47-s − 6.75·49-s − 3.33·51-s + 2.93·53-s + 1.30·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.87·5-s + 0.188·7-s + 0.333·9-s + 0.0936·11-s − 0.0619·13-s − 1.08·15-s + 0.808·17-s + 0.0598·19-s − 0.108·21-s + 1.88·23-s + 2.52·25-s − 0.192·27-s + 0.719·29-s − 0.0592·31-s − 0.0540·33-s + 0.353·35-s − 0.0806·37-s + 0.0357·39-s + 1.50·41-s − 0.357·43-s + 0.626·45-s − 1.45·47-s − 0.964·49-s − 0.466·51-s + 0.403·53-s + 0.175·55-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  =  $(2,\ 8016,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.001414288$
$L(\frac12)$  $\approx$  $3.001414288$
$L(\frac{3}{2})$   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 \( 1 - 4.20T + 5T^{2} \)
7 \( 1 - 0.497T + 7T^{2} \)
11 \( 1 - 0.310T + 11T^{2} \)
13 \( 1 + 0.223T + 13T^{2} \)
17 \( 1 - 3.33T + 17T^{2} \)
19 \( 1 - 0.261T + 19T^{2} \)
23 \( 1 - 9.01T + 23T^{2} \)
29 \( 1 - 3.87T + 29T^{2} \)
31 \( 1 + 0.329T + 31T^{2} \)
37 \( 1 + 0.490T + 37T^{2} \)
41 \( 1 - 9.61T + 41T^{2} \)
43 \( 1 + 2.34T + 43T^{2} \)
47 \( 1 + 9.94T + 47T^{2} \)
53 \( 1 - 2.93T + 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 + 1.23T + 61T^{2} \)
67 \( 1 + 11.6T + 67T^{2} \)
71 \( 1 - 3.61T + 71T^{2} \)
73 \( 1 + 4.33T + 73T^{2} \)
79 \( 1 - 2.59T + 79T^{2} \)
83 \( 1 + 3.06T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 0.603T + 97T^{2} \)
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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

−7.70828895623688615485850578560, −6.93978031799801311309853336996, −6.30931338138105235971483508379, −5.82158974447859152209087654052, −5.04155230277265222702583758840, −4.73390854740518637443054399395, −3.31613469149530862756705783758, −2.60461401564932049345313593296, −1.60995105628986370670513022320, −0.974214997029111080202106368267, 0.974214997029111080202106368267, 1.60995105628986370670513022320, 2.60461401564932049345313593296, 3.31613469149530862756705783758, 4.73390854740518637443054399395, 5.04155230277265222702583758840, 5.82158974447859152209087654052, 6.30931338138105235971483508379, 6.93978031799801311309853336996, 7.70828895623688615485850578560

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