Properties

Label 4-810e2-1.1-c3e2-0-27
Degree $4$
Conductor $656100$
Sign $1$
Analytic cond. $2284.03$
Root an. cond. $6.91314$
Motivic weight $3$
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  + 2·2-s + 5·5-s + 34·7-s − 8·8-s + 10·10-s − 48·11-s + 70·13-s + 68·14-s − 16·16-s + 54·17-s + 238·19-s − 96·22-s − 51·23-s + 140·26-s − 30·29-s + 133·31-s + 108·34-s + 170·35-s + 436·37-s + 476·38-s − 40·40-s + 156·41-s + 88·43-s − 102·46-s − 516·47-s + 343·49-s + 1.27e3·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.447·5-s + 1.83·7-s − 0.353·8-s + 0.316·10-s − 1.31·11-s + 1.49·13-s + 1.29·14-s − 1/4·16-s + 0.770·17-s + 2.87·19-s − 0.930·22-s − 0.462·23-s + 1.05·26-s − 0.192·29-s + 0.770·31-s + 0.544·34-s + 0.821·35-s + 1.93·37-s + 2.03·38-s − 0.158·40-s + 0.594·41-s + 0.312·43-s − 0.326·46-s − 1.60·47-s + 49-s + 3.31·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(656100\)    =    \(2^{2} \cdot 3^{8} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2284.03\)
Root analytic conductor: \(6.91314\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 656100,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(8.732796946\)
\(L(\frac12)\) \(\approx\) \(8.732796946\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - p T + p^{2} T^{2} \)
good7$C_2^2$ \( 1 - 34 T + 813 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 48 T + 973 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 89 T + p^{3} T^{2} )( 1 + 19 T + p^{3} T^{2} ) \)
17$C_2$ \( ( 1 - 27 T + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 119 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 51 T - 9566 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2^2$ \( 1 + 30 T - 23489 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \)
31$C_2^2$ \( 1 - 133 T - 12102 T^{2} - 133 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2$ \( ( 1 - 218 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 156 T - 44585 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} \)
43$C_2^2$ \( 1 - 88 T - 71763 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \)
47$C_2^2$ \( 1 + 516 T + 162433 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2$ \( ( 1 - 639 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 + 654 T + 222337 T^{2} + 654 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 461 T - 14460 T^{2} + 461 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 182 T - 267639 T^{2} + 182 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 900 T + p^{3} T^{2} )^{2} \)
73$C_2$ \( ( 1 - 704 T + p^{3} T^{2} )^{2} \)
79$C_2^2$ \( 1 - 1375 T + 1397586 T^{2} - 1375 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2^2$ \( 1 - 915 T + 265438 T^{2} - 915 p^{3} T^{3} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 1116 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 16 T - 912417 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \)
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

−10.04811115501800610198095171325, −9.754416193856721927091061615111, −9.192371727125854317855212162987, −8.805207847171216179722140953513, −8.192489835755638188117637805286, −7.82700186214416878592518621732, −7.67377028374569141440423726779, −7.28701456072090905241768753497, −6.28341548875111741944558127766, −5.92422470982393828001947677374, −5.66225936490475482568309067677, −5.01006049406086891840836102323, −4.90388750120373026849517716285, −4.37172201179398159031348965997, −3.46051263297646780713331201016, −3.32252180896008198088527012846, −2.50935458184100208720895349576, −1.91814730239620917499931530856, −1.04013047208876226146345326458, −0.898462269197590527335514611004, 0.898462269197590527335514611004, 1.04013047208876226146345326458, 1.91814730239620917499931530856, 2.50935458184100208720895349576, 3.32252180896008198088527012846, 3.46051263297646780713331201016, 4.37172201179398159031348965997, 4.90388750120373026849517716285, 5.01006049406086891840836102323, 5.66225936490475482568309067677, 5.92422470982393828001947677374, 6.28341548875111741944558127766, 7.28701456072090905241768753497, 7.67377028374569141440423726779, 7.82700186214416878592518621732, 8.192489835755638188117637805286, 8.805207847171216179722140953513, 9.192371727125854317855212162987, 9.754416193856721927091061615111, 10.04811115501800610198095171325

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