Properties

Label 4-1058e2-1.1-c1e2-0-1
Degree $4$
Conductor $1119364$
Sign $1$
Analytic cond. $71.3716$
Root an. cond. $2.90657$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s + 3·4-s + 8·6-s + 4·8-s + 6·9-s + 12·12-s + 12·13-s + 5·16-s + 12·18-s + 16·24-s − 10·25-s + 24·26-s − 4·27-s + 12·29-s + 6·32-s + 18·36-s + 48·39-s + 20·48-s − 6·49-s − 20·50-s + 36·52-s − 8·54-s + 24·58-s − 24·59-s + 7·64-s − 24·71-s + ⋯
L(s)  = 1  + 1.41·2-s + 2.30·3-s + 3/2·4-s + 3.26·6-s + 1.41·8-s + 2·9-s + 3.46·12-s + 3.32·13-s + 5/4·16-s + 2.82·18-s + 3.26·24-s − 2·25-s + 4.70·26-s − 0.769·27-s + 2.22·29-s + 1.06·32-s + 3·36-s + 7.68·39-s + 2.88·48-s − 6/7·49-s − 2.82·50-s + 4.99·52-s − 1.08·54-s + 3.15·58-s − 3.12·59-s + 7/8·64-s − 2.84·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1119364 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1119364\)    =    \(2^{2} \cdot 23^{4}\)
Sign: $1$
Analytic conductor: \(71.3716\)
Root analytic conductor: \(2.90657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1119364,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(13.86375469\)
\(L(\frac12)\) \(\approx\) \(13.86375469\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
23 \( 1 \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.3.ae_k
5$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.5.a_k
7$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.7.a_g
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.13.am_ck
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \) 2.19.a_am
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.31.a_ck
37$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.37.a_bq
41$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.41.a_de
43$C_2^2$ \( 1 + 36 T^{2} + p^{2} T^{4} \) 2.43.a_bk
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.53.a_ec
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.61.a_ek
67$C_2^2$ \( 1 + 84 T^{2} + p^{2} T^{4} \) 2.67.a_dg
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 + 150 T^{2} + p^{2} T^{4} \) 2.79.a_fu
83$C_2^2$ \( 1 + 148 T^{2} + p^{2} T^{4} \) 2.83.a_fs
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \) 2.89.a_ge
97$C_2^2$ \( 1 + 192 T^{2} + p^{2} T^{4} \) 2.97.a_hk
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.01066604382513689780746251503, −9.624783590096192512664267577556, −9.108980599533221239758595622788, −8.744372645936980727443534026373, −8.373909592239522074177681012835, −8.091725700805309294701988861238, −7.82907810622905202213209817950, −7.30386566014656959097095013722, −6.42857404741651902896870160008, −6.38119337741399860710805788522, −5.82382202407395136224141531250, −5.54678640604410037737336651077, −4.51051389758033614646582868422, −4.26226595064272013641945891359, −3.60724702281526146069311996944, −3.53377260478980378565524023536, −2.93008994618393166698439108405, −2.65158075281179608546511758965, −1.65032177040719681948808964662, −1.50206886594783616520483378508, 1.50206886594783616520483378508, 1.65032177040719681948808964662, 2.65158075281179608546511758965, 2.93008994618393166698439108405, 3.53377260478980378565524023536, 3.60724702281526146069311996944, 4.26226595064272013641945891359, 4.51051389758033614646582868422, 5.54678640604410037737336651077, 5.82382202407395136224141531250, 6.38119337741399860710805788522, 6.42857404741651902896870160008, 7.30386566014656959097095013722, 7.82907810622905202213209817950, 8.091725700805309294701988861238, 8.373909592239522074177681012835, 8.744372645936980727443534026373, 9.108980599533221239758595622788, 9.624783590096192512664267577556, 10.01066604382513689780746251503

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