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Properties

Label 4-70e4-1.1-c1e2-0-1
Degree $4$
Conductor $24010000$
Sign $1$
Analytic cond. $1530.89$
Root an. cond. $6.25513$
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  + 2·9-s − 8·19-s − 12·29-s + 8·31-s − 12·41-s + 24·59-s − 4·61-s − 24·71-s − 16·79-s − 5·81-s − 12·89-s − 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s − 16·171-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.83·19-s − 2.22·29-s + 1.43·31-s − 1.87·41-s + 3.12·59-s − 0.512·61-s − 2.84·71-s − 1.80·79-s − 5/9·81-s − 1.27·89-s − 1.19·101-s − 0.383·109-s − 2·121-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 + 1.69·169-s − 1.22·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1530.89\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9116679934\)
\(L(\frac12)\) \(\approx\) \(0.9116679934\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
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.m_dq
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.31.ai_da
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$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.47.a_acg
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.53.a_acs
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.59.ay_kc
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.61.e_ew
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.67.a_afa
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.71.y_la
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.73.a_afm
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.305789927634818909714806067556, −8.281609211132587024318060712146, −7.69420788136007504289919218404, −7.33744538383109732752886903906, −6.83390133838138318916790467467, −6.82194443513871435098891095537, −6.39468826572413961008538046613, −5.78948010250637930018609365591, −5.56949570209990711739251984988, −5.27392706656227558858189828497, −4.55029480576120775637440641325, −4.36425479567373422894996999906, −4.01524914012158550274887969545, −3.68732340971917116843635770573, −2.93992618903816531623346940943, −2.76737731920570098049231575658, −1.94109038548430212508388942562, −1.80719762122211448153754426983, −1.19144359295722920959211629250, −0.25942250569838280114146347842, 0.25942250569838280114146347842, 1.19144359295722920959211629250, 1.80719762122211448153754426983, 1.94109038548430212508388942562, 2.76737731920570098049231575658, 2.93992618903816531623346940943, 3.68732340971917116843635770573, 4.01524914012158550274887969545, 4.36425479567373422894996999906, 4.55029480576120775637440641325, 5.27392706656227558858189828497, 5.56949570209990711739251984988, 5.78948010250637930018609365591, 6.39468826572413961008538046613, 6.82194443513871435098891095537, 6.83390133838138318916790467467, 7.33744538383109732752886903906, 7.69420788136007504289919218404, 8.281609211132587024318060712146, 8.305789927634818909714806067556

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