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  =  $(2,\ 96192,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.353661915$
$L(\frac12)$  $\approx$  $2.353661915$
$L(\frac{3}{2})$   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 \( 1 \)
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

−13.47421158569414, −13.26277762066950, −13.15474909859936, −12.32801760498628, −12.00895316571790, −11.32959988510383, −10.67906786144607, −10.14031702726534, −9.849354150837251, −9.557329959127949, −8.942391934527236, −8.315937413418018, −7.664953091135551, −7.226504797969287, −6.689972134398567, −5.977966041442014, −5.711332811305020, −5.246818702905199, −4.540399133259975, −3.686454897541879, −3.163006411591632, −2.686210541584453, −2.179681655599040, −1.125467497111443, −0.5329564594353031, 0.5329564594353031, 1.125467497111443, 2.179681655599040, 2.686210541584453, 3.163006411591632, 3.686454897541879, 4.540399133259975, 5.246818702905199, 5.711332811305020, 5.977966041442014, 6.689972134398567, 7.226504797969287, 7.664953091135551, 8.315937413418018, 8.942391934527236, 9.557329959127949, 9.849354150837251, 10.14031702726534, 10.67906786144607, 11.32959988510383, 12.00895316571790, 12.32801760498628, 13.15474909859936, 13.26277762066950, 13.47421158569414

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