Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 11-s + 6·13-s + 2·17-s + 4·19-s − 25-s − 2·29-s − 8·31-s − 6·37-s + 10·41-s + 4·43-s − 8·47-s + 6·53-s − 2·55-s − 4·59-s − 10·61-s + 12·65-s + 12·67-s − 2·73-s + 16·79-s − 4·83-s + 4·85-s + 18·89-s + 8·95-s − 2·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.301·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.371·29-s − 1.43·31-s − 0.986·37-s + 1.56·41-s + 0.609·43-s − 1.16·47-s + 0.824·53-s − 0.269·55-s − 0.520·59-s − 1.28·61-s + 1.48·65-s + 1.46·67-s − 0.234·73-s + 1.80·79-s − 0.439·83-s + 0.433·85-s + 1.90·89-s + 0.820·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(310464\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{2} \cdot 11\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{310464} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 310464,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
11 \( 1 + T \)
good5 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−13.00499884497537, −12.48816458842068, −11.99910987434560, −11.43041485310886, −10.98179232674014, −10.63489677423806, −10.22886835082586, −9.506482546146187, −9.283846426783245, −8.931497507438614, −8.222503136944642, −7.774863413421354, −7.428056660779366, −6.672034792861041, −6.282555431575030, −5.793622729666419, −5.380594927181793, −5.073469065842915, −4.151676638179647, −3.678559670939124, −3.324422097461272, −2.580796425624429, −2.001224149488846, −1.404548529378219, −0.9809752363552767, 0, 0.9809752363552767, 1.404548529378219, 2.001224149488846, 2.580796425624429, 3.324422097461272, 3.678559670939124, 4.151676638179647, 5.073469065842915, 5.380594927181793, 5.793622729666419, 6.282555431575030, 6.672034792861041, 7.428056660779366, 7.774863413421354, 8.222503136944642, 8.931497507438614, 9.283846426783245, 9.506482546146187, 10.22886835082586, 10.63489677423806, 10.98179232674014, 11.43041485310886, 11.99910987434560, 12.48816458842068, 13.00499884497537

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