Properties

Degree 4
Conductor $ 7^{2} $
Sign $1$
Motivic weight 5
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9·2-s − 6·3-s + 11·4-s − 18·5-s − 54·6-s + 98·7-s − 243·8-s + 54·9-s − 162·10-s + 396·11-s − 66·12-s − 350·13-s + 882·14-s + 108·15-s − 1.38e3·16-s + 1.80e3·17-s + 486·18-s − 3.26e3·19-s − 198·20-s − 588·21-s + 3.56e3·22-s + 2.08e3·23-s + 1.45e3·24-s − 4.58e3·25-s − 3.15e3·26-s − 1.89e3·27-s + 1.07e3·28-s + ⋯
L(s)  = 1  + 1.59·2-s − 0.384·3-s + 0.343·4-s − 0.321·5-s − 0.612·6-s + 0.755·7-s − 1.34·8-s + 2/9·9-s − 0.512·10-s + 0.986·11-s − 0.132·12-s − 0.574·13-s + 1.20·14-s + 0.123·15-s − 1.35·16-s + 1.51·17-s + 0.353·18-s − 2.07·19-s − 0.110·20-s − 0.290·21-s + 1.56·22-s + 0.823·23-s + 0.516·24-s − 1.46·25-s − 0.913·26-s − 0.498·27-s + 0.259·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(49\)    =    \(7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  induced by $\chi_{7} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 49,\ (\ :5/2, 5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.71318\)
\(L(\frac12)\)  \(\approx\)  \(1.71318\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 7$,\(F_p(T)\) is a polynomial of degree 4. If $p = 7$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad7$C_1$ \( ( 1 - p^{2} T )^{2} \)
good2$D_{4}$ \( 1 - 9 T + 35 p T^{2} - 9 p^{5} T^{3} + p^{10} T^{4} \)
3$D_{4}$ \( 1 + 2 p T - 2 p^{2} T^{2} + 2 p^{6} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 18 T + 4906 T^{2} + 18 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 36 p T + 142198 T^{2} - 36 p^{6} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 350 T + 546978 T^{2} + 350 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1800 T + 3567406 T^{2} - 1800 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 3266 T + 7614270 T^{2} + 3266 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2088 T + 9365230 T^{2} - 2088 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 6696 T + 51326470 T^{2} - 6696 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 20 T + 53103102 T^{2} + 20 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 6232 T + 144242070 T^{2} - 6232 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 6048 T + 223864366 T^{2} + 6048 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 3020 T - 30383466 T^{2} + 3020 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 11700 T + 292735582 T^{2} - 11700 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 9468 T + 858185230 T^{2} - 9468 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 43938 T + 1852599934 T^{2} + 43938 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 64754 T + 2408321418 T^{2} + 64754 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 24784 T + 2799959190 T^{2} - 24784 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 97416 T + 5729557966 T^{2} - 97416 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 17452 T + 3828622374 T^{2} - 17452 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 51256 T + 3645565854 T^{2} - 51256 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 117558 T + 7798161502 T^{2} - 117558 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 84276 T + 5915697430 T^{2} - 84276 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 20776 T + 16174049358 T^{2} - 20776 p^{5} T^{3} + p^{10} T^{4} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.76225147842367643539943996651, −21.56672838838674294162066102691, −21.10005866446434830838495013222, −19.81929608229678828534900597185, −19.11442612346681214480330983210, −18.27824021515798302066668506708, −17.17054155342088494212884131690, −16.93896939131145108623869960687, −15.35081808140215523817312865399, −14.82977564781548877745068927308, −14.16894755224873196216556425057, −13.43779131529310651762744374699, −12.20767268553459151312140921785, −12.14564071685321938232185303844, −10.76195289371562611183288647496, −9.368070419283983648535942813665, −8.013647175360602928869050064315, −6.25933100890702232634318684891, −4.95482419936866029068056129325, −3.97604672082062983728337585887, 3.97604672082062983728337585887, 4.95482419936866029068056129325, 6.25933100890702232634318684891, 8.013647175360602928869050064315, 9.368070419283983648535942813665, 10.76195289371562611183288647496, 12.14564071685321938232185303844, 12.20767268553459151312140921785, 13.43779131529310651762744374699, 14.16894755224873196216556425057, 14.82977564781548877745068927308, 15.35081808140215523817312865399, 16.93896939131145108623869960687, 17.17054155342088494212884131690, 18.27824021515798302066668506708, 19.11442612346681214480330983210, 19.81929608229678828534900597185, 21.10005866446434830838495013222, 21.56672838838674294162066102691, 21.76225147842367643539943996651

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