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Properties

Label 4-70e4-1.1-c1e2-0-7
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 − 2·11-s + 6·29-s + 20·31-s + 20·41-s − 20·59-s − 20·61-s + 6·71-s − 26·79-s − 5·81-s + 20·89-s − 4·99-s + 20·101-s − 26·109-s − 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 2/3·9-s − 0.603·11-s + 1.11·29-s + 3.59·31-s + 3.12·41-s − 2.60·59-s − 2.56·61-s + 0.712·71-s − 2.92·79-s − 5/9·81-s + 2.11·89-s − 0.402·99-s + 1.99·101-s − 2.49·109-s − 1.72·121-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 + 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

\(L(1)\) \(\approx\) \(2.683853352\)
\(L(\frac12)\) \(\approx\) \(2.683853352\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.11.c_x
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.13.a_aw
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.17.a_as
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 21 T^{2} + p^{2} T^{4} \) 2.23.a_av
29$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.29.ag_cp
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2^2$ \( 1 - 49 T^{2} + p^{2} T^{4} \) 2.37.a_abx
41$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.41.au_ha
43$C_2^2$ \( 1 - 61 T^{2} + p^{2} T^{4} \) 2.43.a_acj
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.53.a_ag
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.59.u_ik
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2^2$ \( 1 - 109 T^{2} + p^{2} T^{4} \) 2.67.a_aef
71$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.71.ag_fv
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \) 2.79.ba_mp
83$C_2^2$ \( 1 - 66 T^{2} + p^{2} T^{4} \) 2.83.a_aco
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \) 2.97.a_agc
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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

−8.716595911547111699859756072549, −7.915342857876570467368600124693, −7.85590210638802142830123193240, −7.34188038394033501176138032476, −7.19497523797112302082287133162, −6.41449435914160702896619662963, −6.23433495307361781995943467899, −6.15512078619026846115607032073, −5.61263377939723998524460443038, −4.94157223859522518866668280111, −4.68071571226652902646674229920, −4.41523685270143047751841967947, −4.19890571415588465125349760878, −3.44135561016733492216385555604, −2.98034807115167870041253902270, −2.54564755189053783034509103184, −2.48254729751058734828585691274, −1.33550492585439015585516919467, −1.28141161653933366415665582276, −0.48039109127492478095085378225, 0.48039109127492478095085378225, 1.28141161653933366415665582276, 1.33550492585439015585516919467, 2.48254729751058734828585691274, 2.54564755189053783034509103184, 2.98034807115167870041253902270, 3.44135561016733492216385555604, 4.19890571415588465125349760878, 4.41523685270143047751841967947, 4.68071571226652902646674229920, 4.94157223859522518866668280111, 5.61263377939723998524460443038, 6.15512078619026846115607032073, 6.23433495307361781995943467899, 6.41449435914160702896619662963, 7.19497523797112302082287133162, 7.34188038394033501176138032476, 7.85590210638802142830123193240, 7.915342857876570467368600124693, 8.716595911547111699859756072549

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