Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} \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 − 2·5-s + 9-s + 4·11-s − 2·15-s + 4·17-s − 4·19-s + 4·23-s − 25-s + 27-s − 2·29-s + 4·31-s + 4·33-s − 12·37-s − 12·41-s + 8·43-s − 2·45-s + 4·51-s + 14·53-s − 8·55-s − 4·57-s + 2·59-s + 2·61-s + 4·67-s + 4·69-s − 8·71-s − 6·73-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.516·15-s + 0.970·17-s − 0.917·19-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.696·33-s − 1.97·37-s − 1.87·41-s + 1.21·43-s − 0.298·45-s + 0.560·51-s + 1.92·53-s − 1.07·55-s − 0.529·57-s + 0.260·59-s + 0.256·61-s + 0.488·67-s + 0.481·69-s − 0.949·71-s − 0.702·73-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 392784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 392784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392784\)    =    \(2^{4} \cdot 3 \cdot 7^{2} \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{392784} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 392784,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−12.42218960452151, −12.23848909676194, −11.87315506883407, −11.43872744101377, −10.85878414410327, −10.37902859256366, −10.02205893609218, −9.423290363623389, −8.929310429380651, −8.584147472110532, −8.247675078487645, −7.658291368395070, −7.149221974433718, −6.874131323743906, −6.349609288080453, −5.706296615584300, −5.148008309714651, −4.637843052013681, −3.963786155426466, −3.722207525706876, −3.344977195994111, −2.630901394612350, −2.001229082702602, −1.391858932223985, −0.8207135156714110, 0, 0.8207135156714110, 1.391858932223985, 2.001229082702602, 2.630901394612350, 3.344977195994111, 3.722207525706876, 3.963786155426466, 4.637843052013681, 5.148008309714651, 5.706296615584300, 6.349609288080453, 6.874131323743906, 7.149221974433718, 7.658291368395070, 8.247675078487645, 8.584147472110532, 8.929310429380651, 9.423290363623389, 10.02205893609218, 10.37902859256366, 10.85878414410327, 11.43872744101377, 11.87315506883407, 12.23848909676194, 12.42218960452151

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