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Properties

Label 4-1800e2-1.1-c1e2-0-16
Degree $4$
Conductor $3240000$
Sign $1$
Analytic cond. $206.585$
Root an. cond. $3.79118$
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  − 2·2-s + 2·4-s − 4·16-s + 20·31-s + 8·32-s − 8·37-s − 20·41-s − 8·43-s + 10·49-s + 20·53-s − 40·62-s − 8·64-s + 24·67-s + 8·71-s + 16·74-s + 28·79-s + 40·82-s + 16·86-s + 28·89-s − 20·98-s − 40·106-s + 8·107-s + 6·121-s + 40·124-s + 127-s + 131-s − 48·134-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 16-s + 3.59·31-s + 1.41·32-s − 1.31·37-s − 3.12·41-s − 1.21·43-s + 10/7·49-s + 2.74·53-s − 5.08·62-s − 64-s + 2.93·67-s + 0.949·71-s + 1.85·74-s + 3.15·79-s + 4.41·82-s + 1.72·86-s + 2.96·89-s − 2.02·98-s − 3.88·106-s + 0.773·107-s + 6/11·121-s + 3.59·124-s + 0.0887·127-s + 0.0873·131-s − 4.14·134-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3240000\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(206.585\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.150334092\)
\(L(\frac12)\) \(\approx\) \(1.150334092\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.19.a_aw
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.29.a_aw
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.31.au_gg
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.37.i_dm
41$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.41.u_ha
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \) 2.47.a_ada
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.59.a_acc
61$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.61.a_acg
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.67.ay_ks
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.71.ai_gc
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.73.a_abu
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.79.abc_nq
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.89.abc_ok
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \) 2.97.a_adq
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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

−9.504041210953711452785053752026, −8.955750880233050644799054358833, −8.556931648356278760796873431491, −8.483569424605351984240226881498, −8.000734293444069891685454085074, −7.77437763045920858236412704244, −7.03332896996346459011721589276, −6.69941590542705702055565363209, −6.66964371321994701711000277632, −6.09366863299577260243773139382, −5.31685897684416891293912152927, −4.90514381391068282436888010427, −4.82731580885088582726356494480, −3.84606910745892863725119344012, −3.65489764580763467517637586164, −2.93566468355957178072076468554, −2.16886502622145289990205553817, −2.05245409785064956536834752044, −0.997627698734400598114201951715, −0.63920498381537213360836920468, 0.63920498381537213360836920468, 0.997627698734400598114201951715, 2.05245409785064956536834752044, 2.16886502622145289990205553817, 2.93566468355957178072076468554, 3.65489764580763467517637586164, 3.84606910745892863725119344012, 4.82731580885088582726356494480, 4.90514381391068282436888010427, 5.31685897684416891293912152927, 6.09366863299577260243773139382, 6.66964371321994701711000277632, 6.69941590542705702055565363209, 7.03332896996346459011721589276, 7.77437763045920858236412704244, 8.000734293444069891685454085074, 8.483569424605351984240226881498, 8.556931648356278760796873431491, 8.955750880233050644799054358833, 9.504041210953711452785053752026

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