Properties

Label 4-39e4-1.1-c3e2-0-4
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $8053.60$
Root an. cond. $9.47322$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 7·4-s − 15·5-s − 15·7-s + 15·8-s + 75·10-s + 17·11-s + 75·14-s − 65·16-s − 70·17-s + 141·19-s − 105·20-s − 85·22-s − 145·23-s + 25·25-s − 105·28-s − 34·29-s + 140·31-s + 95·32-s + 350·34-s + 225·35-s + 190·37-s − 705·38-s − 225·40-s + 538·41-s + 455·43-s + 119·44-s + ⋯
L(s)  = 1  − 1.76·2-s + 7/8·4-s − 1.34·5-s − 0.809·7-s + 0.662·8-s + 2.37·10-s + 0.465·11-s + 1.43·14-s − 1.01·16-s − 0.998·17-s + 1.70·19-s − 1.17·20-s − 0.823·22-s − 1.31·23-s + 1/5·25-s − 0.708·28-s − 0.217·29-s + 0.811·31-s + 0.524·32-s + 1.76·34-s + 1.08·35-s + 0.844·37-s − 3.00·38-s − 0.889·40-s + 2.04·41-s + 1.61·43-s + 0.407·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(8053.60\)
Root analytic conductor: \(9.47322\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1521} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2313441,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$D_{4}$ \( 1 + 5 T + 9 p T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 3 p T + 8 p^{2} T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \)
7$D_{4}$ \( 1 + 15 T + 738 T^{2} + 15 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 17 T + 1778 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 70 T + 9963 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 141 T + 978 p T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 145 T + 24962 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 34 T + 33767 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 140 T + 21982 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 190 T + 102103 T^{2} - 190 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 538 T + 208503 T^{2} - 538 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 455 T + 170782 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 60 T + 125246 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 545 T + 256304 T^{2} + 545 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 809 T + 561522 T^{2} + 809 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 502 T + 346963 T^{2} - 502 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 475 T + 622738 T^{2} - 475 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 127 T + 672998 T^{2} - 127 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 585 T + 832884 T^{2} + 585 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 240 T + 993678 T^{2} - 240 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 260 T + 1117974 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 921 T + 1555592 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 415 T + 933568 T^{2} - 415 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.028846860301285114815939076465, −8.492553806430241363141624930796, −8.040624311993847007093685666135, −7.895243051377449855482698446461, −7.40662131209223513604549297313, −7.38098639561304497483699160589, −6.45576384966031824968973947886, −6.31181112906464042209212185517, −5.82723221069999670170110774503, −5.13101321923083465182337117206, −4.37589210240560850595647254091, −4.36407617482326653610571373176, −3.74906813297077226033666012331, −3.25886754824020089737239112160, −2.70143500491642066022922190434, −2.03329052862266446654586087612, −1.14480537131054448136326617302, −0.848997921979859223264229607217, 0, 0, 0.848997921979859223264229607217, 1.14480537131054448136326617302, 2.03329052862266446654586087612, 2.70143500491642066022922190434, 3.25886754824020089737239112160, 3.74906813297077226033666012331, 4.36407617482326653610571373176, 4.37589210240560850595647254091, 5.13101321923083465182337117206, 5.82723221069999670170110774503, 6.31181112906464042209212185517, 6.45576384966031824968973947886, 7.38098639561304497483699160589, 7.40662131209223513604549297313, 7.895243051377449855482698446461, 8.040624311993847007093685666135, 8.492553806430241363141624930796, 9.028846860301285114815939076465

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