Properties

Label 4-462e2-1.1-c5e2-0-7
Degree $4$
Conductor $213444$
Sign $1$
Analytic cond. $5490.41$
Root an. cond. $8.60798$
Motivic weight $5$
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  + 8·2-s + 18·3-s + 48·4-s − 94·5-s + 144·6-s + 98·7-s + 256·8-s + 243·9-s − 752·10-s + 242·11-s + 864·12-s − 1.02e3·13-s + 784·14-s − 1.69e3·15-s + 1.28e3·16-s − 1.19e3·17-s + 1.94e3·18-s − 3.71e3·19-s − 4.51e3·20-s + 1.76e3·21-s + 1.93e3·22-s − 5.59e3·23-s + 4.60e3·24-s + 1.51e3·25-s − 8.17e3·26-s + 2.91e3·27-s + 4.70e3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.68·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s − 2.37·10-s + 0.603·11-s + 1.73·12-s − 1.67·13-s + 1.06·14-s − 1.94·15-s + 5/4·16-s − 1.00·17-s + 1.41·18-s − 2.36·19-s − 2.52·20-s + 0.872·21-s + 0.852·22-s − 2.20·23-s + 1.63·24-s + 0.484·25-s − 2.37·26-s + 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 213444 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(213444\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(5490.41\)
Root analytic conductor: \(8.60798\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 213444,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
3$C_1$ \( ( 1 - p^{2} T )^{2} \)
7$C_1$ \( ( 1 - p^{2} T )^{2} \)
11$C_1$ \( ( 1 - p^{2} T )^{2} \)
good5$D_{4}$ \( 1 + 94 T + 7323 T^{2} + 94 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 1022 T + 931003 T^{2} + 1022 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1192 T + 1422486 T^{2} + 1192 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 3718 T + 8019283 T^{2} + 3718 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 5596 T + 17918006 T^{2} + 5596 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 3546 T + 12862211 T^{2} - 3546 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 440 T - 28291182 T^{2} + 440 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 - 7826 T + 42706699 T^{2} - 7826 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 2096 T + 156606690 T^{2} + 2096 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 11220 T + 130312542 T^{2} - 11220 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 26962 T + 626049875 T^{2} + 26962 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 1036 T - 22191054 T^{2} + 1036 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 29738 T + 1477706835 T^{2} + 29738 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 41964 T + 2018144142 T^{2} + 41964 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 33526 T + 1430162787 T^{2} + 33526 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 29840 T + 2326091166 T^{2} + 29840 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 136594 T + 8790528691 T^{2} + 136594 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 121320 T + 8106700094 T^{2} + 121320 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 6256 T + 6826702794 T^{2} - 6256 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 72628 T + 5778859094 T^{2} + 72628 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 150740 T + 12214791370 T^{2} + 150740 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.952690505988120652218521942373, −9.822315096442543220830833227538, −8.697213947999816145820220671682, −8.683491754786894752469491051591, −8.009471435778760430102113133435, −7.75472316619842323206318978099, −7.30940952693105037335873687706, −6.92190621410061718337400818209, −6.11281274557531399529595660552, −5.97374656145148064304937754503, −4.63648610219975742839346808770, −4.53129150694689960471845291775, −4.24782760894609214013543422859, −3.95534272348918927898561766662, −3.05080762904590884214881056588, −2.65810549799792901996717267531, −1.92036280061107086663919245764, −1.66712437009284723471416694151, 0, 0, 1.66712437009284723471416694151, 1.92036280061107086663919245764, 2.65810549799792901996717267531, 3.05080762904590884214881056588, 3.95534272348918927898561766662, 4.24782760894609214013543422859, 4.53129150694689960471845291775, 4.63648610219975742839346808770, 5.97374656145148064304937754503, 6.11281274557531399529595660552, 6.92190621410061718337400818209, 7.30940952693105037335873687706, 7.75472316619842323206318978099, 8.009471435778760430102113133435, 8.683491754786894752469491051591, 8.697213947999816145820220671682, 9.822315096442543220830833227538, 9.952690505988120652218521942373

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