Properties

Label 4-35e4-1.1-c3e2-0-5
Degree $4$
Conductor $1500625$
Sign $1$
Analytic cond. $5224.01$
Root an. cond. $8.50160$
Motivic weight $3$
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  − 8·2-s + 2·3-s + 34·4-s − 16·6-s − 96·8-s − 19·9-s − 14·11-s + 68·12-s + 50·13-s + 196·16-s − 50·17-s + 152·18-s − 36·19-s + 112·22-s − 244·23-s − 192·24-s − 400·26-s − 30·27-s − 26·29-s + 120·31-s − 352·32-s − 28·33-s + 400·34-s − 646·36-s − 564·37-s + 288·38-s + 100·39-s + ⋯
L(s)  = 1  − 2.82·2-s + 0.384·3-s + 17/4·4-s − 1.08·6-s − 4.24·8-s − 0.703·9-s − 0.383·11-s + 1.63·12-s + 1.06·13-s + 3.06·16-s − 0.713·17-s + 1.99·18-s − 0.434·19-s + 1.08·22-s − 2.21·23-s − 1.63·24-s − 3.01·26-s − 0.213·27-s − 0.166·29-s + 0.695·31-s − 1.94·32-s − 0.147·33-s + 2.01·34-s − 2.99·36-s − 2.50·37-s + 1.22·38-s + 0.410·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1500625\)    =    \(5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(5224.01\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1225} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1500625,\ (\ :3/2, 3/2),\ 1)\)

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)$
bad5 \( 1 \)
7 \( 1 \)
good2$D_{4}$ \( 1 + p^{3} T + 15 p T^{2} + p^{6} T^{3} + p^{6} T^{4} \)
3$D_{4}$ \( 1 - 2 T + 23 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 14 T + 663 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 50 T + 4987 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 50 T + 387 p T^{2} + 50 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 36 T + 10170 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 244 T + 29970 T^{2} + 244 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 26 T + 47795 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 120 T - 1618 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 564 T + 173630 T^{2} + 564 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 8 p T + 133986 T^{2} - 8 p^{4} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 260 T + 166666 T^{2} - 260 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 350 T + 203423 T^{2} + 350 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 56 T + 265770 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 616 T + p^{3} T^{2} )^{2} \)
61$D_{4}$ \( 1 + 336 T + 458858 T^{2} + 336 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 152 T + 599110 T^{2} - 152 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 952 T + p^{3} T^{2} )^{2} \)
73$D_{4}$ \( 1 - 676 T + 655606 T^{2} - 676 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 1014 T + 1120119 T^{2} - 1014 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 376 T + 458918 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 216 T + 1417730 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 2742 T + 3608187 T^{2} - 2742 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.063735970955052681629463298847, −8.753121347542266478128498846534, −8.342058649674969757797721330066, −8.299694689327958777216252248981, −7.64428176531600296330580043408, −7.56593629459955471466369014225, −6.82529093893812924891762575321, −6.51287040492505255272973767710, −6.01292167170788064406480949044, −5.63948132721623645014725336614, −4.98641766015974544172455919274, −4.16956198183376017704801919054, −3.76025736354589034692134286515, −3.14854441715333259544207396108, −2.29693089291660890722972924719, −2.16547646323359179056678802407, −1.47822155308891159926625694764, −0.872637716312671756744814455202, 0, 0, 0.872637716312671756744814455202, 1.47822155308891159926625694764, 2.16547646323359179056678802407, 2.29693089291660890722972924719, 3.14854441715333259544207396108, 3.76025736354589034692134286515, 4.16956198183376017704801919054, 4.98641766015974544172455919274, 5.63948132721623645014725336614, 6.01292167170788064406480949044, 6.51287040492505255272973767710, 6.82529093893812924891762575321, 7.56593629459955471466369014225, 7.64428176531600296330580043408, 8.299694689327958777216252248981, 8.342058649674969757797721330066, 8.753121347542266478128498846534, 9.063735970955052681629463298847

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