↑

Properties

Label 4-2624e2-1.1-c1e2-0-1
Degree $4$
Conductor $6885376$
Sign $1$
Analytic cond. $439.017$
Root an. cond. $4.57741$
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  + 4·9-s − 10·25-s − 16·31-s − 16·37-s − 6·41-s − 8·43-s − 4·49-s − 24·59-s − 4·61-s − 8·73-s + 7·81-s + 24·83-s + 32·103-s − 24·107-s + 24·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + ⋯
L(s)  = 1  + 4/3·9-s − 2·25-s − 2.87·31-s − 2.63·37-s − 0.937·41-s − 1.21·43-s − 4/7·49-s − 3.12·59-s − 0.512·61-s − 0.936·73-s + 7/9·81-s + 2.63·83-s + 3.15·103-s − 2.32·107-s + 2.25·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2·169-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6885376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(6885376\)    =    \(2^{12} \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(439.017\)
Root analytic conductor: \(4.57741\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 6885376,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9545395945\)
\(L(\frac12)\) \(\approx\) \(0.9545395945\)
\(L(\frac{3}{2})\) 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 \( 1 \)
41$C_2$ \( 1 + 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.3.a_ae
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.7.a_e
11$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.11.a_au
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.17.a_ac
19$C_2^2$ \( 1 - 20 T^{2} + p^{2} T^{4} \) 2.19.a_au
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 26 T^{2} + p^{2} T^{4} \) 2.29.a_aba
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.37.q_fi
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.47.a_abs
53$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \) 2.53.a_acw
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.67.a_bc
71$C_2^2$ \( 1 - 92 T^{2} + p^{2} T^{4} \) 2.71.a_ado
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.73.i_gg
79$C_2^2$ \( 1 - 140 T^{2} + p^{2} T^{4} \) 2.79.a_afk
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.a_afq
97$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.97.a_dq
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

−8.985134151867115640523445142303, −8.866075227866857550836819381075, −8.265164950013336515653916636109, −7.70081597186566041226450183921, −7.60803725656428188240656341310, −7.24010372352042687131081395235, −6.79778644123537583222602567557, −6.46853478469199689626870616326, −5.94341082922476311366489680634, −5.58549645048709757072919856356, −5.11587721543301209089770674059, −4.73856801935362261965124357049, −4.34065754937014080256948385059, −3.72887269355348827472378854782, −3.39467125770590920931157172520, −3.23706577260673729092840060627, −1.99358704231421364497479188934, −1.79657262909454826687669165540, −1.61005578564380831285356289469, −0.30252267618718571088788787267, 0.30252267618718571088788787267, 1.61005578564380831285356289469, 1.79657262909454826687669165540, 1.99358704231421364497479188934, 3.23706577260673729092840060627, 3.39467125770590920931157172520, 3.72887269355348827472378854782, 4.34065754937014080256948385059, 4.73856801935362261965124357049, 5.11587721543301209089770674059, 5.58549645048709757072919856356, 5.94341082922476311366489680634, 6.46853478469199689626870616326, 6.79778644123537583222602567557, 7.24010372352042687131081395235, 7.60803725656428188240656341310, 7.70081597186566041226450183921, 8.265164950013336515653916636109, 8.866075227866857550836819381075, 8.985134151867115640523445142303

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