Properties

Label 4-3920e2-1.1-c1e2-0-11
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·5-s + 3·9-s − 7·11-s − 3·13-s − 2·15-s − 5·17-s + 2·19-s − 2·23-s + 3·25-s − 8·27-s − 3·29-s − 16·31-s + 7·33-s − 4·37-s + 3·39-s − 2·41-s + 6·43-s + 6·45-s + 3·47-s + 5·51-s + 10·53-s − 14·55-s − 2·57-s + 16·59-s − 6·61-s − 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 9-s − 2.11·11-s − 0.832·13-s − 0.516·15-s − 1.21·17-s + 0.458·19-s − 0.417·23-s + 3/5·25-s − 1.53·27-s − 0.557·29-s − 2.87·31-s + 1.21·33-s − 0.657·37-s + 0.480·39-s − 0.312·41-s + 0.914·43-s + 0.894·45-s + 0.437·47-s + 0.700·51-s + 1.37·53-s − 1.88·55-s − 0.264·57-s + 2.08·59-s − 0.768·61-s − 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 50 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 10 T + 98 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 6 T + 98 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 13 T + 126 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 18 T + 226 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 8 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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

−8.221345665559606901660482760792, −7.84203182632699118213000383556, −7.38608807688222561539267122752, −7.11751261256207745575014903405, −7.05156195854645885923483017149, −6.49490272768468117134143742294, −5.74576289682242546573333651805, −5.62918632422261289477208296107, −5.36873739608471021665711810954, −5.17887727183785227383142718214, −4.49511260737420677607847196876, −4.17016369837727554170397268357, −3.74053866015848729635518726844, −3.13160720684153040146763666133, −2.49426885531844182262658675487, −2.17012654594016786844304897496, −1.93401038058346366834480938811, −1.17694073576242688207767912336, 0, 0, 1.17694073576242688207767912336, 1.93401038058346366834480938811, 2.17012654594016786844304897496, 2.49426885531844182262658675487, 3.13160720684153040146763666133, 3.74053866015848729635518726844, 4.17016369837727554170397268357, 4.49511260737420677607847196876, 5.17887727183785227383142718214, 5.36873739608471021665711810954, 5.62918632422261289477208296107, 5.74576289682242546573333651805, 6.49490272768468117134143742294, 7.05156195854645885923483017149, 7.11751261256207745575014903405, 7.38608807688222561539267122752, 7.84203182632699118213000383556, 8.221345665559606901660482760792

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