Properties

Label 4-1008e2-1.1-c5e2-0-7
Degree $4$
Conductor $1016064$
Sign $1$
Analytic cond. $26136.1$
Root an. cond. $12.7148$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·5-s − 98·7-s + 54·11-s + 240·13-s − 1.90e3·17-s + 400·19-s + 2.93e3·23-s − 1.89e3·25-s − 5.54e3·29-s − 4.90e3·31-s − 980·35-s + 2.95e3·37-s − 6.07e3·41-s − 3.30e3·43-s + 1.69e4·47-s + 7.20e3·49-s − 6.89e3·53-s + 540·55-s + 5.38e4·59-s + 4.14e3·61-s + 2.40e3·65-s − 3.35e4·67-s + 7.35e4·71-s + 128·73-s − 5.29e3·77-s − 5.77e4·79-s + 2.30e4·83-s + ⋯
L(s)  = 1  + 0.178·5-s − 0.755·7-s + 0.134·11-s + 0.393·13-s − 1.59·17-s + 0.254·19-s + 1.15·23-s − 0.606·25-s − 1.22·29-s − 0.916·31-s − 0.135·35-s + 0.354·37-s − 0.563·41-s − 0.272·43-s + 1.12·47-s + 3/7·49-s − 0.337·53-s + 0.0240·55-s + 2.01·59-s + 0.142·61-s + 0.0704·65-s − 0.912·67-s + 1.73·71-s + 0.00281·73-s − 0.101·77-s − 1.04·79-s + 0.366·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1016064\)    =    \(2^{8} \cdot 3^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(26136.1\)
Root analytic conductor: \(12.7148\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1016064,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(2.894588645\)
\(L(\frac12)\) \(\approx\) \(2.894588645\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7$C_1$ \( ( 1 + p^{2} T )^{2} \)
good5$D_{4}$ \( 1 - 2 p T + 1994 T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 54 T + 284302 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 - 240 T + 739862 T^{2} - 240 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1906 T + 1860002 T^{2} + 1906 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 400 T + 3279798 T^{2} - 400 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2930 T + 9774686 T^{2} - 2930 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 + 5540 T + 40407182 T^{2} + 5540 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 4904 T + 59914302 T^{2} + 4904 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 2952 T + 138794486 T^{2} - 2952 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 6070 T + 232254602 T^{2} + 6070 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 3304 T + 197837766 T^{2} + 3304 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 16988 T + 429309854 T^{2} - 16988 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 6896 T + 613869254 T^{2} + 6896 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 53820 T + 1699029142 T^{2} - 53820 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 68 p T + 1287381294 T^{2} - 68 p^{6} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 33516 T + 2261444678 T^{2} + 33516 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 73518 T + 4934300734 T^{2} - 73518 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 128 T - 360358674 T^{2} - 128 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 57740 T + 6120687198 T^{2} + 57740 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 23016 T - 4303421546 T^{2} - 23016 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 141530 T + 14809201898 T^{2} - 141530 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 226216 T + 29691907182 T^{2} + 226216 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.254587649706692344170659737037, −9.143425804005610720596668849924, −8.708529799704381972108460860644, −8.341978885992136615692489784352, −7.64386010879464661436241738590, −7.32732889653923515081195896568, −6.73664207031382757215138944125, −6.71617997821547441167861923240, −5.95354460019274356443735578834, −5.70956035145050303436706763992, −5.14446048769584502440866744108, −4.69745102408123532946109220437, −3.90800501490904085827915141929, −3.86973685056225665922856852398, −3.14484314405418520022129797794, −2.66281851428337519919138822866, −1.96817828290049143746193664862, −1.72296216793397064908415001792, −0.67483672312843545708990423496, −0.46722074126278856589504561030, 0.46722074126278856589504561030, 0.67483672312843545708990423496, 1.72296216793397064908415001792, 1.96817828290049143746193664862, 2.66281851428337519919138822866, 3.14484314405418520022129797794, 3.86973685056225665922856852398, 3.90800501490904085827915141929, 4.69745102408123532946109220437, 5.14446048769584502440866744108, 5.70956035145050303436706763992, 5.95354460019274356443735578834, 6.71617997821547441167861923240, 6.73664207031382757215138944125, 7.32732889653923515081195896568, 7.64386010879464661436241738590, 8.341978885992136615692489784352, 8.708529799704381972108460860644, 9.143425804005610720596668849924, 9.254587649706692344170659737037

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