Properties

Degree 2
Conductor $ 2^{4} \cdot 5 \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  − 2·3-s − 5-s − 4·7-s + 9-s − 11-s − 4·13-s + 2·15-s + 4·19-s + 8·21-s − 6·23-s + 25-s + 4·27-s + 6·29-s + 8·31-s + 2·33-s + 4·35-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s − 45-s − 6·47-s + 9·49-s − 6·53-s + 55-s − 8·57-s + 12·59-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 0.917·19-s + 1.74·21-s − 1.25·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.43·31-s + 0.348·33-s + 0.676·35-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.824·53-s + 0.134·55-s − 1.05·57-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(254320\)    =    \(2^{4} \cdot 5 \cdot 11 \cdot 17^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{254320} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 254320,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(0.4468811347\)
\(L(\frac12)\)  \(\approx\)  \(0.4468811347\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;11,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;11,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
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

−12.50931985833896, −12.28177785933943, −12.04106231529824, −11.58025890306475, −11.08014809788472, −10.42853049106105, −9.972535342762937, −9.856733751900997, −9.384781604558488, −8.414719143428514, −8.248623891290479, −7.614075172238138, −6.868942930399780, −6.630505327079310, −6.365655973568613, −5.612039424789607, −5.173784132723349, −4.828086401717747, −4.133521269895492, −3.539797421471300, −2.923023396366990, −2.640888686801831, −1.688700457385729, −0.7607588112039889, −0.2772653736476592, 0.2772653736476592, 0.7607588112039889, 1.688700457385729, 2.640888686801831, 2.923023396366990, 3.539797421471300, 4.133521269895492, 4.828086401717747, 5.173784132723349, 5.612039424789607, 6.365655973568613, 6.630505327079310, 6.868942930399780, 7.614075172238138, 8.248623891290479, 8.414719143428514, 9.384781604558488, 9.856733751900997, 9.972535342762937, 10.42853049106105, 11.08014809788472, 11.58025890306475, 12.04106231529824, 12.28177785933943, 12.50931985833896

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