Properties

Label 4-2205e2-1.1-c1e2-0-14
Degree $4$
Conductor $4862025$
Sign $1$
Analytic cond. $310.006$
Root an. cond. $4.19607$
Motivic weight $1$
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  + 3·4-s + 2·5-s + 12·11-s + 5·16-s − 12·19-s + 6·20-s − 25-s − 4·29-s + 20·31-s + 4·41-s + 36·44-s + 24·55-s + 16·59-s + 4·61-s + 3·64-s − 20·71-s − 36·76-s − 8·79-s + 10·80-s − 12·89-s − 24·95-s − 3·100-s − 12·101-s − 4·109-s − 12·116-s + 86·121-s + 60·124-s + ⋯
L(s)  = 1  + 3/2·4-s + 0.894·5-s + 3.61·11-s + 5/4·16-s − 2.75·19-s + 1.34·20-s − 1/5·25-s − 0.742·29-s + 3.59·31-s + 0.624·41-s + 5.42·44-s + 3.23·55-s + 2.08·59-s + 0.512·61-s + 3/8·64-s − 2.37·71-s − 4.12·76-s − 0.900·79-s + 1.11·80-s − 1.27·89-s − 2.46·95-s − 0.299·100-s − 1.19·101-s − 0.383·109-s − 1.11·116-s + 7.81·121-s + 5.38·124-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4862025\)    =    \(3^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(310.006\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4862025,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.420668986\)
\(L(\frac12)\) \(\approx\) \(6.420668986\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
5$C_2$ \( 1 - 2 T + p T^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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

−9.152015619096937725207148203158, −8.926832819636393698701746971037, −8.412928316515727147529402649335, −8.324086314088260903286211992260, −7.61913039043795394897084926451, −6.94374272381862301057236724196, −6.74289647431043218547640321919, −6.59765281391358485624757472496, −6.27469358068490926207003117890, −5.86039425343849187090355705307, −5.70687740042207680765704750497, −4.49555301422447838119620216232, −4.43550523097462089520048091438, −4.00056205547625356527093407031, −3.55116177053758679806291577369, −2.68955168677024235883511577316, −2.48261377113560904590093539749, −1.75625004967303957429671382069, −1.54357849409318562023686122849, −0.910232915939716783487750563057, 0.910232915939716783487750563057, 1.54357849409318562023686122849, 1.75625004967303957429671382069, 2.48261377113560904590093539749, 2.68955168677024235883511577316, 3.55116177053758679806291577369, 4.00056205547625356527093407031, 4.43550523097462089520048091438, 4.49555301422447838119620216232, 5.70687740042207680765704750497, 5.86039425343849187090355705307, 6.27469358068490926207003117890, 6.59765281391358485624757472496, 6.74289647431043218547640321919, 6.94374272381862301057236724196, 7.61913039043795394897084926451, 8.324086314088260903286211992260, 8.412928316515727147529402649335, 8.926832819636393698701746971037, 9.152015619096937725207148203158

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