Properties

Degree $4$
Conductor $518400$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 5-s + 6·9-s − 5·11-s + 3·15-s + 6·17-s − 10·19-s + 6·23-s + 9·27-s + 10·29-s − 2·31-s − 15·33-s + 8·37-s + 3·41-s + 3·43-s + 6·45-s + 4·47-s + 7·49-s + 18·51-s − 12·53-s − 5·55-s − 30·57-s − 3·59-s − 2·61-s − 11·67-s + 18·69-s + 28·71-s + ⋯
L(s)  = 1  + 1.73·3-s + 0.447·5-s + 2·9-s − 1.50·11-s + 0.774·15-s + 1.45·17-s − 2.29·19-s + 1.25·23-s + 1.73·27-s + 1.85·29-s − 0.359·31-s − 2.61·33-s + 1.31·37-s + 0.468·41-s + 0.457·43-s + 0.894·45-s + 0.583·47-s + 49-s + 2.52·51-s − 1.64·53-s − 0.674·55-s − 3.97·57-s − 0.390·59-s − 0.256·61-s − 1.34·67-s + 2.16·69-s + 3.32·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 518400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(518400\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{720} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 518400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.95014\)
\(L(\frac12)\) \(\approx\) \(3.95014\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
good7$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 3 T - 34 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 15 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.30167182590907834215945763363, −10.26020690497743039708356183530, −9.841654842723544589284375624556, −9.025303657793254598679494721108, −9.012308300768752647510164394594, −8.575926059178326469351972147858, −7.87465109184261690182734328560, −7.82352083783939644058943236720, −7.44372152204783509840973182660, −6.74371997742015913801222992217, −6.18736440487051839131936384388, −5.86297783752437163521518093388, −4.97085517556857291745178790127, −4.69898396479728600753142460449, −4.13254258241428061702997466158, −3.39268277321198310993092640098, −2.89651450367539122506627930403, −2.50226890156502896041659072926, −1.98991972099260763786963171994, −0.977723197946596028285680891564, 0.977723197946596028285680891564, 1.98991972099260763786963171994, 2.50226890156502896041659072926, 2.89651450367539122506627930403, 3.39268277321198310993092640098, 4.13254258241428061702997466158, 4.69898396479728600753142460449, 4.97085517556857291745178790127, 5.86297783752437163521518093388, 6.18736440487051839131936384388, 6.74371997742015913801222992217, 7.44372152204783509840973182660, 7.82352083783939644058943236720, 7.87465109184261690182734328560, 8.575926059178326469351972147858, 9.012308300768752647510164394594, 9.025303657793254598679494721108, 9.841654842723544589284375624556, 10.26020690497743039708356183530, 10.30167182590907834215945763363

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