Properties

Label 4-5400e2-1.1-c1e2-0-13
Degree $4$
Conductor $29160000$
Sign $1$
Analytic cond. $1859.26$
Root an. cond. $6.56652$
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  + 8·11-s + 2·19-s − 8·31-s + 5·49-s − 8·59-s − 10·61-s + 16·71-s + 10·79-s − 24·89-s + 28·109-s + 26·121-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 + 197-s + ⋯
L(s)  = 1  + 2.41·11-s + 0.458·19-s − 1.43·31-s + 5/7·49-s − 1.04·59-s − 1.28·61-s + 1.89·71-s + 1.12·79-s − 2.54·89-s + 2.68·109-s + 2.36·121-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 + 0.0712·197-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(29160000\)    =    \(2^{6} \cdot 3^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1859.26\)
Root analytic conductor: \(6.56652\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.915586079\)
\(L(\frac12)\) \(\approx\) \(2.915586079\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 169 T^{2} + p^{2} T^{4} \)
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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

−8.470360958805363914906805967017, −7.79069649305554665643431146012, −7.76541664297207504092583255487, −7.21496693114564215195257678505, −6.84075251870433879531768741192, −6.66682891041724654249548330039, −6.23677329590089121039487044327, −5.84210667852036916964007291831, −5.62242918571371554028641927105, −4.92855318238906594259106760175, −4.80208071255630176286821099423, −4.14910827577004919924162215779, −3.87509137148009522121856517899, −3.59748960480907163482422588339, −3.19696358568573385484819134056, −2.56688594706573486144994342202, −2.07461257374055932086993191387, −1.37354839828689688535446946434, −1.34570797303715441464810617132, −0.46952281390110531252636695270, 0.46952281390110531252636695270, 1.34570797303715441464810617132, 1.37354839828689688535446946434, 2.07461257374055932086993191387, 2.56688594706573486144994342202, 3.19696358568573385484819134056, 3.59748960480907163482422588339, 3.87509137148009522121856517899, 4.14910827577004919924162215779, 4.80208071255630176286821099423, 4.92855318238906594259106760175, 5.62242918571371554028641927105, 5.84210667852036916964007291831, 6.23677329590089121039487044327, 6.66682891041724654249548330039, 6.84075251870433879531768741192, 7.21496693114564215195257678505, 7.76541664297207504092583255487, 7.79069649305554665643431146012, 8.470360958805363914906805967017

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