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Properties

Label 4-5400e2-1.1-c1e2-0-7
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  + 10·11-s − 4·19-s − 12·29-s − 14·31-s − 12·41-s + 5·49-s + 8·59-s − 16·61-s − 16·71-s − 32·79-s − 12·89-s + 18·101-s − 20·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  + 3.01·11-s − 0.917·19-s − 2.22·29-s − 2.51·31-s − 1.87·41-s + 5/7·49-s + 1.04·59-s − 2.04·61-s − 1.89·71-s − 3.60·79-s − 1.27·89-s + 1.79·101-s − 1.91·109-s + 4.81·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 29160000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.158005052\)
\(L(\frac12)\) \(\approx\) \(1.158005052\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.7.a_af
11$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \) 2.11.ak_bv
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.a_ak
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.a_be
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \) 2.23.a_abq
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.29.m_dq
31$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \) 2.31.o_eh
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.m_eo
43$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \) 2.43.a_ade
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2^2$ \( 1 - 81 T^{2} + p^{2} T^{4} \) 2.53.a_add
59$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.59.ai_fe
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.61.q_he
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.71.q_hy
73$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \) 2.73.a_afp
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.79.bg_py
83$C_2^2$ \( 1 - 45 T^{2} + p^{2} T^{4} \) 2.83.a_abt
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 193 T^{2} + p^{2} T^{4} \) 2.97.a_ahl
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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

−8.615495351766044192084741097264, −7.920959637946580340140411278262, −7.50564367765200221301608759911, −7.21787666304516195158655795546, −6.98979661228171531634185494352, −6.50783527736378785509248610138, −6.36206863660346596386242591381, −5.68238009373076973321178143258, −5.67854238736469912943297810963, −5.21520433357114184351626430047, −4.41420557126733876977935607197, −4.23797212329524637497636238290, −3.98694790468124806575689375957, −3.47778947356838967625892409343, −3.32617298361677094913514036588, −2.58043159623514795103542932919, −1.80504140097477334791153832179, −1.60253438084587106743806142977, −1.41053815764737135214798380682, −0.27195242167192809556675305017, 0.27195242167192809556675305017, 1.41053815764737135214798380682, 1.60253438084587106743806142977, 1.80504140097477334791153832179, 2.58043159623514795103542932919, 3.32617298361677094913514036588, 3.47778947356838967625892409343, 3.98694790468124806575689375957, 4.23797212329524637497636238290, 4.41420557126733876977935607197, 5.21520433357114184351626430047, 5.67854238736469912943297810963, 5.68238009373076973321178143258, 6.36206863660346596386242591381, 6.50783527736378785509248610138, 6.98979661228171531634185494352, 7.21787666304516195158655795546, 7.50564367765200221301608759911, 7.920959637946580340140411278262, 8.615495351766044192084741097264

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