Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 11 \cdot 17^{2} $
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 − 2·5-s − 2·7-s + 9-s + 11-s − 2·13-s − 2·15-s + 6·19-s − 2·21-s − 25-s + 27-s + 8·29-s − 8·31-s + 33-s + 4·35-s − 10·37-s − 2·39-s − 8·41-s + 2·43-s − 2·45-s + 8·47-s − 3·49-s − 2·53-s − 2·55-s + 6·57-s − 12·59-s − 10·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s + 1.37·19-s − 0.436·21-s − 1/5·25-s + 0.192·27-s + 1.48·29-s − 1.43·31-s + 0.174·33-s + 0.676·35-s − 1.64·37-s − 0.320·39-s − 1.24·41-s + 0.304·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.269·55-s + 0.794·57-s − 1.56·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(152592\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{152592} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 152592,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7373664176$
$L(\frac12)$  $\approx$  $0.7373664176$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.46600058610562, −12.70272657782919, −12.30882307062038, −12.04750441242809, −11.56570670819715, −10.89520546724838, −10.40279252975233, −9.937708182859233, −9.366500838730921, −9.093676494750909, −8.445213707668454, −8.001503194949057, −7.420475669545389, −7.056669601314498, −6.709126426914493, −5.820960484928214, −5.452515525395722, −4.611767151662831, −4.310428340939662, −3.486301684137878, −3.221859231741962, −2.792127670988567, −1.814326907837550, −1.286094546776305, −0.2446007710895278, 0.2446007710895278, 1.286094546776305, 1.814326907837550, 2.792127670988567, 3.221859231741962, 3.486301684137878, 4.310428340939662, 4.611767151662831, 5.452515525395722, 5.820960484928214, 6.709126426914493, 7.056669601314498, 7.420475669545389, 8.001503194949057, 8.445213707668454, 9.093676494750909, 9.366500838730921, 9.937708182859233, 10.40279252975233, 10.89520546724838, 11.56570670819715, 12.04750441242809, 12.30882307062038, 12.70272657782919, 13.46600058610562

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