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Properties

Label 4-2700e2-1.1-c1e2-0-8
Degree $4$
Conductor $7290000$
Sign $1$
Analytic cond. $464.816$
Root an. cond. $4.64323$
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  − 12·11-s + 2·19-s + 12·29-s + 16·31-s + 12·41-s + 13·49-s + 22·61-s + 12·71-s + 2·79-s − 24·89-s + 20·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯
L(s)  = 1  − 3.61·11-s + 0.458·19-s + 2.22·29-s + 2.87·31-s + 1.87·41-s + 13/7·49-s + 2.81·61-s + 1.42·71-s + 0.225·79-s − 2.54·89-s + 1.91·109-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(7290000\)    =    \(2^{4} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(464.816\)
Root analytic conductor: \(4.64323\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 7290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.237350495\)
\(L(\frac12)\) \(\approx\) \(2.237350495\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.7.a_an
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.11.m_cg
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.13.a_az
17$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.17.a_abi
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.19.ac_bn
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.23.a_ak
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.31.aq_ew
37$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.37.a_az
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$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.47.a_by
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \) 2.61.aw_jj
67$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.67.a_adh
71$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.71.am_gw
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.79.ac_gd
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.89.y_mk
97$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.97.a_az
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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

−8.662005344780482857512107138103, −8.646447319931435621723778486705, −8.228750818772929242373553556010, −7.950961017954233406849241021547, −7.50436432220991161132668150533, −7.40183345547681027639334664554, −6.55668090600298430594086124511, −6.54642944371342397416588372841, −5.76888189222842868587102579353, −5.54597927496260557904653656806, −4.99934506431491502306299490311, −4.98562726913011504771304098781, −4.30199016195112327218552354374, −4.00440381354498784722132274492, −2.99734575086180618444286341179, −2.80069967900779266725902995997, −2.57853459821101907334827209308, −2.14203024117508226005110013508, −0.854476625427570910611834974982, −0.65951577567304629641748861218, 0.65951577567304629641748861218, 0.854476625427570910611834974982, 2.14203024117508226005110013508, 2.57853459821101907334827209308, 2.80069967900779266725902995997, 2.99734575086180618444286341179, 4.00440381354498784722132274492, 4.30199016195112327218552354374, 4.98562726913011504771304098781, 4.99934506431491502306299490311, 5.54597927496260557904653656806, 5.76888189222842868587102579353, 6.54642944371342397416588372841, 6.55668090600298430594086124511, 7.40183345547681027639334664554, 7.50436432220991161132668150533, 7.950961017954233406849241021547, 8.228750818772929242373553556010, 8.646447319931435621723778486705, 8.662005344780482857512107138103

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