↑

Properties

Label 4-70e4-1.1-c1e2-0-6
Degree $4$
Conductor $24010000$
Sign $1$
Analytic cond. $1530.89$
Root an. cond. $6.25513$
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  + 5·9-s − 2·11-s + 12·19-s − 6·29-s + 4·31-s − 20·41-s − 28·59-s − 16·61-s + 22·79-s + 16·81-s − 8·89-s − 10·99-s − 20·101-s + 22·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 60·171-s + 173-s + ⋯
L(s)  = 1  + 5/3·9-s − 0.603·11-s + 2.75·19-s − 1.11·29-s + 0.718·31-s − 3.12·41-s − 3.64·59-s − 2.04·61-s + 2.47·79-s + 16/9·81-s − 0.847·89-s − 1.00·99-s − 1.99·101-s + 2.10·109-s − 1.72·121-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 + 1/13·169-s + 4.58·171-s + 0.0760·173-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1530.89\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.683169982\)
\(L(\frac12)\) \(\approx\) \(2.683169982\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
11$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.13.a_ab
17$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \) 2.17.a_abh
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.29.g_cp
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.47.a_abt
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.53.a_ady
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.59.bc_mc
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.67.a_ck
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.79.aw_kt
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \) 2.83.a_afu
89$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.89.i_hm
97$C_2^2$ \( 1 - 185 T^{2} + p^{2} T^{4} \) 2.97.a_ahd
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

−8.242135337932297305403067708443, −7.902442672220571329731694647512, −7.64142340022396011646497228486, −7.57871120688647191177386278863, −6.96801728605908224285553901697, −6.72826854177635356416922438819, −6.43410110139514074923901713489, −5.80436373880083372645068435375, −5.47331316701825921266528869724, −5.12049585700382583389213360473, −4.71891267453636074536423633706, −4.53994739523548076393664159338, −3.89398435673951544669968263752, −3.47687900923214761571225487748, −3.00064520257618873930694319209, −2.97091408801980420023771552852, −1.84540430148615545707932662814, −1.69151683799610837406794011830, −1.24136756422567987164184934698, −0.45940019415673966553784621328, 0.45940019415673966553784621328, 1.24136756422567987164184934698, 1.69151683799610837406794011830, 1.84540430148615545707932662814, 2.97091408801980420023771552852, 3.00064520257618873930694319209, 3.47687900923214761571225487748, 3.89398435673951544669968263752, 4.53994739523548076393664159338, 4.71891267453636074536423633706, 5.12049585700382583389213360473, 5.47331316701825921266528869724, 5.80436373880083372645068435375, 6.43410110139514074923901713489, 6.72826854177635356416922438819, 6.96801728605908224285553901697, 7.57871120688647191177386278863, 7.64142340022396011646497228486, 7.902442672220571329731694647512, 8.242135337932297305403067708443

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