↑

Properties

Label 4-650e2-1.1-c1e2-0-8
Degree $4$
Conductor $422500$
Sign $1$
Analytic cond. $26.9389$
Root an. cond. $2.27821$
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  − 2·2-s + 3·4-s − 6·7-s − 4·8-s + 5·9-s + 6·13-s + 12·14-s + 5·16-s − 10·18-s − 12·26-s − 18·28-s − 6·32-s + 15·36-s − 6·37-s − 6·47-s + 13·49-s + 18·52-s + 24·56-s − 16·61-s − 30·63-s + 7·64-s + 24·67-s − 20·72-s + 12·73-s + 12·74-s − 20·79-s + 16·81-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 2.26·7-s − 1.41·8-s + 5/3·9-s + 1.66·13-s + 3.20·14-s + 5/4·16-s − 2.35·18-s − 2.35·26-s − 3.40·28-s − 1.06·32-s + 5/2·36-s − 0.986·37-s − 0.875·47-s + 13/7·49-s + 2.49·52-s + 3.20·56-s − 2.04·61-s − 3.77·63-s + 7/8·64-s + 2.93·67-s − 2.35·72-s + 1.40·73-s + 1.39·74-s − 2.25·79-s + 16/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 422500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(422500\)    =    \(2^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(26.9389\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 422500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7571139293\)
\(L(\frac12)\) \(\approx\) \(0.7571139293\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
5 \( 1 \)
13$C_2$ \( 1 - 6 T + p T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
7$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.7.g_x
11$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.11.a_aw
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.19.a_ac
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.37.g_df
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.43.a_adh
47$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.47.g_dz
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.59.a_ade
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \) 2.71.a_df
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.79.u_jy
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.83.am_hu
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.89.a_afm
97$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.97.ay_na
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

−10.40579359910264910391798374226, −10.19309663544502108599700135739, −9.993688555409133294724844030653, −9.382608289109218655888987767139, −9.244139125722055831391721374611, −8.726277275654772402775989839184, −8.289418765404660343800427367972, −7.60815704598803955156863689976, −7.37227262827544519118519171246, −6.65967277115333027023738769657, −6.43008308137672900757969774338, −6.34249403667160232528714015235, −5.56859466857697651659723215691, −4.77996152384419800370966864288, −3.95306942499852943931287753441, −3.40462859029285778796014360296, −3.26775815526395431429032431597, −2.18130367835354612418786059287, −1.47953711797463536056081496378, −0.62604777101715275959992429842, 0.62604777101715275959992429842, 1.47953711797463536056081496378, 2.18130367835354612418786059287, 3.26775815526395431429032431597, 3.40462859029285778796014360296, 3.95306942499852943931287753441, 4.77996152384419800370966864288, 5.56859466857697651659723215691, 6.34249403667160232528714015235, 6.43008308137672900757969774338, 6.65967277115333027023738769657, 7.37227262827544519118519171246, 7.60815704598803955156863689976, 8.289418765404660343800427367972, 8.726277275654772402775989839184, 9.244139125722055831391721374611, 9.382608289109218655888987767139, 9.993688555409133294724844030653, 10.19309663544502108599700135739, 10.40579359910264910391798374226

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