↑

Properties

Label 4-2340e2-1.1-c1e2-0-19
Degree $4$
Conductor $5475600$
Sign $1$
Analytic cond. $349.129$
Root an. cond. $4.32261$
Motivic weight $1$
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  + 4·5-s − 8·7-s − 4·11-s − 6·13-s + 6·17-s + 4·19-s + 4·23-s + 11·25-s − 32·35-s − 8·37-s − 14·41-s − 16·43-s − 8·47-s + 34·49-s − 14·53-s − 16·55-s − 20·59-s − 24·65-s − 16·71-s + 32·77-s − 32·83-s + 24·85-s + 14·89-s + 48·91-s + 16·95-s + 12·103-s − 24·107-s + ⋯
L(s)  = 1  + 1.78·5-s − 3.02·7-s − 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.917·19-s + 0.834·23-s + 11/5·25-s − 5.40·35-s − 1.31·37-s − 2.18·41-s − 2.43·43-s − 1.16·47-s + 34/7·49-s − 1.92·53-s − 2.15·55-s − 2.60·59-s − 2.97·65-s − 1.89·71-s + 3.64·77-s − 3.51·83-s + 2.60·85-s + 1.48·89-s + 5.03·91-s + 1.64·95-s + 1.18·103-s − 2.32·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5475600\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(349.129\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 5475600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + 6 T + p T^{2} \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.7.i_be
11$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.11.e_i
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ag_s
19$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.19.ae_i
23$C_2^2$ \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) 2.23.ae_i
29$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.29.a_acg
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.41.o_du
43$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.43.q_ey
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.47.i_eg
53$C_2^2$ \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) 2.53.o_du
59$C_2^2$ \( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} \) 2.59.u_hs
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2^2$ \( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} \) 2.71.q_ey
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.83.bg_qg
89$C_2^2$ \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) 2.89.ao_du
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.97.a_aby
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

−9.027074940449329767547232396564, −8.592418190755540312639416511759, −7.898275587303619777552339454912, −7.60978486535125028838948782598, −6.99329622354881268789926514028, −6.84464896533915457673292007035, −6.49361173029117896295245680310, −6.13451951016493620510389181412, −5.53450002656236067441915310135, −5.45101542749872187963752009334, −4.89493559679637309286733959321, −4.71715910682561326586774216120, −3.40730708516430712594009986859, −3.27823051428770651340401643859, −2.88847212116849012526576496884, −2.86467461739435012030812574016, −1.82920143743077499597703524280, −1.45068759366909346772990994053, 0, 0, 1.45068759366909346772990994053, 1.82920143743077499597703524280, 2.86467461739435012030812574016, 2.88847212116849012526576496884, 3.27823051428770651340401643859, 3.40730708516430712594009986859, 4.71715910682561326586774216120, 4.89493559679637309286733959321, 5.45101542749872187963752009334, 5.53450002656236067441915310135, 6.13451951016493620510389181412, 6.49361173029117896295245680310, 6.84464896533915457673292007035, 6.99329622354881268789926514028, 7.60978486535125028838948782598, 7.898275587303619777552339454912, 8.592418190755540312639416511759, 9.027074940449329767547232396564

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