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Properties

Label 4-2100e2-1.1-c1e2-0-13
Degree $4$
Conductor $4410000$
Sign $1$
Analytic cond. $281.185$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 9-s + 4·11-s + 4·19-s + 4·29-s − 12·31-s + 12·41-s − 49-s − 20·61-s − 28·71-s + 16·79-s + 81-s + 28·89-s − 4·99-s − 12·101-s − 4·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s − 4·171-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.20·11-s + 0.917·19-s + 0.742·29-s − 2.15·31-s + 1.87·41-s − 1/7·49-s − 2.56·61-s − 3.32·71-s + 1.80·79-s + 1/9·81-s + 2.96·89-s − 0.402·99-s − 1.19·101-s − 0.383·109-s − 0.909·121-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 + 0.769·169-s − 0.305·171-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4410000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(281.185\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4410000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.190797893\)
\(L(\frac12)\) \(\approx\) \(2.190797893\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.11.ae_ba
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.19.ae_bq
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.31.m_du
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \) 2.37.a_abm
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.41.am_eo
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.53.a_abq
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.67.a_k
71$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.71.bc_na
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.79.aq_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−9.144593807832547590411866122455, −8.996631486353940443550737405198, −8.815961489863402958343030494724, −7.913947825464193298343188648119, −7.79096866954398236662650254631, −7.42835712050021839219411176940, −7.01724407870329187713910140161, −6.47127456969836905409128006463, −6.14750204015814133902886614277, −5.84212625150835119824347244565, −5.33156950713950827684392588529, −4.86500582160276436637647678343, −4.46044355146702724611483695480, −3.81679617722819951054387158047, −3.67820661039193682650115059487, −2.93294280468274022841632116872, −2.65150396854639750385185438266, −1.72292186223764928279766773872, −1.41814933429407969278404174820, −0.54326065222918145208496200978, 0.54326065222918145208496200978, 1.41814933429407969278404174820, 1.72292186223764928279766773872, 2.65150396854639750385185438266, 2.93294280468274022841632116872, 3.67820661039193682650115059487, 3.81679617722819951054387158047, 4.46044355146702724611483695480, 4.86500582160276436637647678343, 5.33156950713950827684392588529, 5.84212625150835119824347244565, 6.14750204015814133902886614277, 6.47127456969836905409128006463, 7.01724407870329187713910140161, 7.42835712050021839219411176940, 7.79096866954398236662650254631, 7.913947825464193298343188648119, 8.815961489863402958343030494724, 8.996631486353940443550737405198, 9.144593807832547590411866122455

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