Properties

Label 4-5400e2-1.1-c1e2-0-9
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

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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

\(L(1)\) \(\approx\) \(1.273420042\)
\(L(\frac12)\) \(\approx\) \(1.273420042\)
\(L(\frac{3}{2})\) 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

−8.086592132677921025191161548005, −8.041764823393839272873227907413, −7.64908150247277608003320668676, −7.37024241375507521877431021002, −7.06303904397808918582503602544, −6.63199738985647254933871627640, −6.06042880226455265433924219883, −5.75271000851018500666721532981, −5.46253275237421675685337453950, −5.19506992395321585401884981296, −4.59033382765709404948124791056, −4.57715545261719796054605469987, −3.83120707306438466663407756064, −3.34208327373457238836778933890, −3.09438724081690456112133826257, −2.63210187062134322044630878622, −2.01611707602557258234476057677, −1.92280245962109758082073944973, −0.921132347323204340622513395384, −0.34392919915695631528291515964, 0.34392919915695631528291515964, 0.921132347323204340622513395384, 1.92280245962109758082073944973, 2.01611707602557258234476057677, 2.63210187062134322044630878622, 3.09438724081690456112133826257, 3.34208327373457238836778933890, 3.83120707306438466663407756064, 4.57715545261719796054605469987, 4.59033382765709404948124791056, 5.19506992395321585401884981296, 5.46253275237421675685337453950, 5.75271000851018500666721532981, 6.06042880226455265433924219883, 6.63199738985647254933871627640, 7.06303904397808918582503602544, 7.37024241375507521877431021002, 7.64908150247277608003320668676, 8.041764823393839272873227907413, 8.086592132677921025191161548005

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