Properties

Label 4-3520e2-1.1-c1e2-0-16
Degree $4$
Conductor $12390400$
Sign $1$
Analytic cond. $790.022$
Root an. cond. $5.30163$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 8·7-s + 4·9-s − 6·11-s + 4·19-s + 3·25-s + 16·35-s + 20·37-s + 16·43-s + 8·45-s + 34·49-s − 12·53-s − 12·55-s + 32·63-s − 48·77-s − 4·79-s + 7·81-s + 24·83-s + 8·95-s + 4·97-s − 24·99-s − 36·113-s + 25·121-s + 4·125-s + 127-s + 131-s + 32·133-s + ⋯
L(s)  = 1  + 0.894·5-s + 3.02·7-s + 4/3·9-s − 1.80·11-s + 0.917·19-s + 3/5·25-s + 2.70·35-s + 3.28·37-s + 2.43·43-s + 1.19·45-s + 34/7·49-s − 1.64·53-s − 1.61·55-s + 4.03·63-s − 5.47·77-s − 0.450·79-s + 7/9·81-s + 2.63·83-s + 0.820·95-s + 0.406·97-s − 2.41·99-s − 3.38·113-s + 2.27·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 2.77·133-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(12390400\)    =    \(2^{12} \cdot 5^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(790.022\)
Root analytic conductor: \(5.30163\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 12390400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.922566033\)
\(L(\frac12)\) \(\approx\) \(6.922566033\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
11$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 32 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 116 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 128 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.567612284534629396987333877814, −8.288260031874163880767609762372, −7.914746697882272378203167037103, −7.64166451244791050360745819229, −7.38327534739755925917224078977, −7.35150165484839559474439481210, −6.26828534734889237879066566675, −6.20702295797552933127415202685, −5.62403091291961679091445895181, −5.18992454454424185534833903778, −4.97021835673624518191213741892, −4.72901747300712654311857826796, −4.23441240359332023763871535362, −4.02117387198327979129427338191, −3.01987943396996280184289890032, −2.58295526097096036518957154949, −2.17036935034886081771008460732, −1.81676006985515900704494677727, −1.10377402522156738563092725633, −0.963610871386801885959524970623, 0.963610871386801885959524970623, 1.10377402522156738563092725633, 1.81676006985515900704494677727, 2.17036935034886081771008460732, 2.58295526097096036518957154949, 3.01987943396996280184289890032, 4.02117387198327979129427338191, 4.23441240359332023763871535362, 4.72901747300712654311857826796, 4.97021835673624518191213741892, 5.18992454454424185534833903778, 5.62403091291961679091445895181, 6.20702295797552933127415202685, 6.26828534734889237879066566675, 7.35150165484839559474439481210, 7.38327534739755925917224078977, 7.64166451244791050360745819229, 7.914746697882272378203167037103, 8.288260031874163880767609762372, 8.567612284534629396987333877814

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