Properties

Label 4-1932e2-1.1-c1e2-0-9
Degree $4$
Conductor $3732624$
Sign $1$
Analytic cond. $237.995$
Root an. cond. $3.92773$
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  + 2·3-s − 5-s − 2·7-s + 3·9-s − 6·11-s − 13-s − 2·15-s − 4·19-s − 4·21-s − 2·23-s − 8·25-s + 4·27-s − 8·29-s − 6·31-s − 12·33-s + 2·35-s + 10·37-s − 2·39-s − 6·41-s − 7·43-s − 3·45-s − 14·47-s + 3·49-s − 3·53-s + 6·55-s − 8·57-s − 17·59-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s − 0.755·7-s + 9-s − 1.80·11-s − 0.277·13-s − 0.516·15-s − 0.917·19-s − 0.872·21-s − 0.417·23-s − 8/5·25-s + 0.769·27-s − 1.48·29-s − 1.07·31-s − 2.08·33-s + 0.338·35-s + 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.06·43-s − 0.447·45-s − 2.04·47-s + 3/7·49-s − 0.412·53-s + 0.809·55-s − 1.05·57-s − 2.21·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3732624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3732624\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(237.995\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1932} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 3732624,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
3$C_1$ \( ( 1 - T )^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 + T + 25 T^{2} + p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 8 T + 69 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 10 T + 79 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 7 T + 67 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 17 T + 189 T^{2} + 17 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 117 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 7 T + 45 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 7 T + 153 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 157 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 113 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 9 T + 47 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
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

−9.015146915657015209417224531927, −8.558956470719536268682309433030, −8.049486843936738470746652085513, −7.929392683743312878378190862746, −7.48670368017283528488447982189, −7.40100983509761279854631995234, −6.50827852513291290224241119764, −6.45474482146570766627574299893, −5.61881713229407850963289442402, −5.56887439675100329374204240796, −4.64351345236886376271401914771, −4.60536603581385114101347172527, −3.75763484064554852632384062720, −3.60005369286613938794179311800, −3.01344737718619690504388969003, −2.64326968259702339137451269125, −1.94770729116318433966066273669, −1.71866424722829054788062165324, 0, 0, 1.71866424722829054788062165324, 1.94770729116318433966066273669, 2.64326968259702339137451269125, 3.01344737718619690504388969003, 3.60005369286613938794179311800, 3.75763484064554852632384062720, 4.60536603581385114101347172527, 4.64351345236886376271401914771, 5.56887439675100329374204240796, 5.61881713229407850963289442402, 6.45474482146570766627574299893, 6.50827852513291290224241119764, 7.40100983509761279854631995234, 7.48670368017283528488447982189, 7.929392683743312878378190862746, 8.049486843936738470746652085513, 8.558956470719536268682309433030, 9.015146915657015209417224531927

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