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Properties

Label 4-3900e2-1.1-c1e2-0-11
Degree $4$
Conductor $15210000$
Sign $1$
Analytic cond. $969.802$
Root an. cond. $5.58047$
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  − 9-s − 4·19-s + 12·29-s + 4·31-s − 24·41-s + 10·49-s − 24·59-s + 4·61-s + 24·71-s − 16·79-s + 81-s + 36·101-s − 28·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + 4·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 1/3·9-s − 0.917·19-s + 2.22·29-s + 0.718·31-s − 3.74·41-s + 10/7·49-s − 3.12·59-s + 0.512·61-s + 2.84·71-s − 1.80·79-s + 1/9·81-s + 3.58·101-s − 2.68·109-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 − 0.0769·169-s + 0.305·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15210000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(969.802\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 15210000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.497747924\)
\(L(\frac12)\) \(\approx\) \(1.497747924\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.7.a_ak
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.17.a_c
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.23.a_abu
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.29.am_dq
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.31.ae_co
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.41.y_is
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.43.a_acs
47$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.47.a_adq
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.y_kc
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.67.a_abi
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.71.ay_la
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.79.q_io
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.83.a_aw
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
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

−8.553393549374940667645947902979, −8.246338351633735368616140169139, −8.223191789834371459505444545092, −7.53165228627309250225968932222, −7.16816886494807905424675747520, −6.68962048994562177913917311785, −6.51625251273592792889649360143, −6.17292887377036500962934671750, −5.74835771598664034187146743665, −5.18605028454802285187354187023, −4.79871073124165505586241629979, −4.67888752918136377747521631973, −4.10226210043198031200649336447, −3.50090845177034762887597751417, −3.29640906426465562293523460796, −2.66293715557982258690257735159, −2.35187024738839499729743681622, −1.68629817374324296152238761595, −1.18003313679719617666703663629, −0.37211195649736004729770519061, 0.37211195649736004729770519061, 1.18003313679719617666703663629, 1.68629817374324296152238761595, 2.35187024738839499729743681622, 2.66293715557982258690257735159, 3.29640906426465562293523460796, 3.50090845177034762887597751417, 4.10226210043198031200649336447, 4.67888752918136377747521631973, 4.79871073124165505586241629979, 5.18605028454802285187354187023, 5.74835771598664034187146743665, 6.17292887377036500962934671750, 6.51625251273592792889649360143, 6.68962048994562177913917311785, 7.16816886494807905424675747520, 7.53165228627309250225968932222, 8.223191789834371459505444545092, 8.246338351633735368616140169139, 8.553393549374940667645947902979

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