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Properties

Label 4-2100e2-1.1-c1e2-0-14
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 + 12·11-s − 4·19-s − 12·29-s − 20·31-s − 12·41-s − 49-s + 28·61-s + 12·71-s + 32·79-s + 81-s − 12·89-s − 12·99-s + 12·101-s − 4·109-s + 86·121-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 + 3.61·11-s − 0.917·19-s − 2.22·29-s − 3.59·31-s − 1.87·41-s − 1/7·49-s + 3.58·61-s + 1.42·71-s + 3.60·79-s + 1/9·81-s − 1.27·89-s − 1.20·99-s + 1.19·101-s − 0.383·109-s + 7.81·121-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

\(L(1)\) \(\approx\) \(2.329139561\)
\(L(\frac12)\) \(\approx\) \(2.329139561\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.11.am_cg
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.31.u_gg
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_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$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \) 2.53.a_bm
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.61.abc_mg
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \) 2.67.a_aeo
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.79.abg_py
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \) 2.97.a_ck
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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

−9.155355237928304641317401512772, −9.056905264667100048224988577378, −8.646693035088994163307033674817, −8.324123597042410467535872598423, −7.64120898165077936271498657367, −7.29533866146785588398474346426, −6.72968007592384811071474199259, −6.70070818044907245779431159408, −6.34479701767229900640850277198, −5.65456109008119805397148376645, −5.42627908550052787580229016327, −4.97811158927648025340578562254, −4.13010747690806675939735742786, −3.85423576287053218733809732813, −3.63597240633513617544941809826, −3.42843394096223263262900199321, −2.05370804337113063915514180621, −1.99769552241843360414095510257, −1.45130304828502986774155297989, −0.54789257247745583718774084444, 0.54789257247745583718774084444, 1.45130304828502986774155297989, 1.99769552241843360414095510257, 2.05370804337113063915514180621, 3.42843394096223263262900199321, 3.63597240633513617544941809826, 3.85423576287053218733809732813, 4.13010747690806675939735742786, 4.97811158927648025340578562254, 5.42627908550052787580229016327, 5.65456109008119805397148376645, 6.34479701767229900640850277198, 6.70070818044907245779431159408, 6.72968007592384811071474199259, 7.29533866146785588398474346426, 7.64120898165077936271498657367, 8.324123597042410467535872598423, 8.646693035088994163307033674817, 9.056905264667100048224988577378, 9.155355237928304641317401512772

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