Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \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  + 2·5-s + 4·7-s + 4·11-s + 4·17-s − 4·19-s − 4·23-s − 25-s − 2·29-s − 4·31-s + 8·35-s + 12·37-s − 12·41-s − 8·43-s + 9·49-s + 14·53-s + 8·55-s − 2·59-s + 2·61-s − 4·67-s + 8·71-s + 6·73-s + 16·77-s + 14·79-s − 6·83-s + 8·85-s + 6·89-s − 8·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s + 1.20·11-s + 0.970·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.718·31-s + 1.35·35-s + 1.97·37-s − 1.87·41-s − 1.21·43-s + 9/7·49-s + 1.92·53-s + 1.07·55-s − 0.260·59-s + 0.256·61-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 1.82·77-s + 1.57·79-s − 0.658·83-s + 0.867·85-s + 0.635·89-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(96192\)    =    \(2^{6} \cdot 3^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{96192} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 96192,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.668584857\)
\(L(\frac12)\)  \(\approx\)  \(4.668584857\)
\(L(\frac{3}{2})\)   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 \( 1 \)
167 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
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

−14.05217404850891, −13.30456260669197, −12.96819415452018, −12.15127252306911, −11.76253938274509, −11.49445834716840, −10.84968524725261, −10.28070846943881, −9.873128231939351, −9.371365623042123, −8.750514439249378, −8.339438236333670, −7.835488263674539, −7.294437586999138, −6.599024123067588, −6.088750585612564, −5.658635268785121, −5.023808083311824, −4.568852658458224, −3.842460795553047, −3.458615268377478, −2.271857673927788, −1.965834095767415, −1.439909587053892, −0.6935030435929477, 0.6935030435929477, 1.439909587053892, 1.965834095767415, 2.271857673927788, 3.458615268377478, 3.842460795553047, 4.568852658458224, 5.023808083311824, 5.658635268785121, 6.088750585612564, 6.599024123067588, 7.294437586999138, 7.835488263674539, 8.339438236333670, 8.750514439249378, 9.371365623042123, 9.873128231939351, 10.28070846943881, 10.84968524725261, 11.49445834716840, 11.76253938274509, 12.15127252306911, 12.96819415452018, 13.30456260669197, 14.05217404850891

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