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Properties

Label 4-340e2-1.1-c1e2-0-11
Degree $4$
Conductor $115600$
Sign $1$
Analytic cond. $7.37075$
Root an. cond. $1.64769$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 4·5-s + 6·9-s + 8·10-s − 10·13-s − 4·16-s + 2·17-s + 12·18-s + 8·20-s + 11·25-s − 20·26-s + 6·29-s − 8·32-s + 4·34-s + 12·36-s − 18·41-s + 24·45-s + 14·49-s + 22·50-s − 20·52-s + 10·53-s + 12·58-s − 2·61-s − 8·64-s − 40·65-s + 4·68-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 1.78·5-s + 2·9-s + 2.52·10-s − 2.77·13-s − 16-s + 0.485·17-s + 2.82·18-s + 1.78·20-s + 11/5·25-s − 3.92·26-s + 1.11·29-s − 1.41·32-s + 0.685·34-s + 2·36-s − 2.81·41-s + 3.57·45-s + 2·49-s + 3.11·50-s − 2.77·52-s + 1.37·53-s + 1.57·58-s − 0.256·61-s − 64-s − 4.96·65-s + 0.485·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(115600\)    =    \(2^{4} \cdot 5^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(7.37075\)
Root analytic conductor: \(1.64769\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 115600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.528555605\)
\(L(\frac12)\) \(\approx\) \(4.528555605\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
17$C_2$ \( 1 - 2 T + p T^{2} \)
good3$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.3.a_ag
7$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.7.a_ao
11$C_2^2$ \( 1 + p^{2} T^{4} \) 2.11.a_a
13$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.13.k_by
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.29.ag_s
31$C_2^2$ \( 1 + p^{2} T^{4} \) 2.31.a_a
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.37.a_cs
41$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.s_gg
43$C_2^2$ \( 1 + p^{2} T^{4} \) 2.43.a_a
47$C_2^2$ \( 1 + p^{2} T^{4} \) 2.47.a_a
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.53.ak_by
59$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.59.a_aeo
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.c_c
67$C_2^2$ \( 1 + p^{2} T^{4} \) 2.67.a_a
71$C_2^2$ \( 1 + p^{2} T^{4} \) 2.71.a_a
73$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \) 2.73.bg_pm
79$C_2^2$ \( 1 + p^{2} T^{4} \) 2.79.a_a
83$C_2^2$ \( 1 + p^{2} T^{4} \) 2.83.a_a
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.89.a_ada
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.a_afa
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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

−12.21865394629368533184480610926, −11.69431373428469232181807142007, −10.61457869549460501572686219746, −10.08438922984808083876834747440, −10.05579916893730978642107876863, −9.894617985312893775373921855865, −8.933504899168652619661294375062, −8.839342633340608358796285506442, −7.58590952706957762838451471840, −7.18857007012695378858377860522, −6.96954690119841036969558048922, −6.34249079861059558868101757174, −5.79533000359576900647538228805, −5.03612993325664126270226211589, −4.96073291594366195872110422111, −4.46957914806854771484456401131, −3.64140717969400077208659671047, −2.68893858100097238984954980283, −2.31027450024773564066264643361, −1.50088273099326648897544736405, 1.50088273099326648897544736405, 2.31027450024773564066264643361, 2.68893858100097238984954980283, 3.64140717969400077208659671047, 4.46957914806854771484456401131, 4.96073291594366195872110422111, 5.03612993325664126270226211589, 5.79533000359576900647538228805, 6.34249079861059558868101757174, 6.96954690119841036969558048922, 7.18857007012695378858377860522, 7.58590952706957762838451471840, 8.839342633340608358796285506442, 8.933504899168652619661294375062, 9.894617985312893775373921855865, 10.05579916893730978642107876863, 10.08438922984808083876834747440, 10.61457869549460501572686219746, 11.69431373428469232181807142007, 12.21865394629368533184480610926

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