↑

Properties

Label 4-4400e2-1.1-c1e2-0-1
Degree $4$
Conductor $19360000$
Sign $1$
Analytic cond. $1234.41$
Root an. cond. $5.92740$
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  + 6·9-s + 2·11-s − 16·19-s − 20·29-s − 16·31-s − 4·41-s + 10·49-s − 8·59-s + 20·61-s − 8·79-s + 27·81-s − 12·89-s + 12·99-s + 28·101-s + 4·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s − 96·171-s + ⋯
L(s)  = 1  + 2·9-s + 0.603·11-s − 3.67·19-s − 3.71·29-s − 2.87·31-s − 0.624·41-s + 10/7·49-s − 1.04·59-s + 2.56·61-s − 0.900·79-s + 3·81-s − 1.27·89-s + 1.20·99-s + 2.78·101-s + 0.383·109-s + 3/11·121-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 − 7.34·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(19360000\)    =    \(2^{8} \cdot 5^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1234.41\)
Root analytic conductor: \(5.92740\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 19360000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7431391575\)
\(L(\frac12)\) \(\approx\) \(0.7431391575\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
13$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.13.a_aba
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.19.q_dy
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.23.a_s
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.29.u_gc
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.31.q_ew
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.37.a_ba
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.41.e_di
43$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.43.a_aby
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.47.a_abe
53$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.a_dm
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.59.i_fe
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.73.a_ade
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.79.i_gs
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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

−8.643537153037795994690228853748, −8.233534680830904802576332703701, −7.60653758125453381460823461758, −7.29753960001246101262495865526, −7.28353569542483018708836282952, −6.73015654793833406618220027928, −6.50012029258434021074980432500, −5.93741964595754633704130381082, −5.70844906895903716326390942239, −5.20787547662399056070055629642, −4.74486459160823184004287558951, −4.20353733430988535345696739498, −3.99192640734493392743961311446, −3.76062417333265384353872895755, −3.49575347005544356926014876528, −2.32129383064925947799850511398, −2.03512810024324411219103687305, −1.86237394473426923529474242763, −1.32205053360290164875469187472, −0.22942666032940754763124551861, 0.22942666032940754763124551861, 1.32205053360290164875469187472, 1.86237394473426923529474242763, 2.03512810024324411219103687305, 2.32129383064925947799850511398, 3.49575347005544356926014876528, 3.76062417333265384353872895755, 3.99192640734493392743961311446, 4.20353733430988535345696739498, 4.74486459160823184004287558951, 5.20787547662399056070055629642, 5.70844906895903716326390942239, 5.93741964595754633704130381082, 6.50012029258434021074980432500, 6.73015654793833406618220027928, 7.28353569542483018708836282952, 7.29753960001246101262495865526, 7.60653758125453381460823461758, 8.233534680830904802576332703701, 8.643537153037795994690228853748

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