Properties

Label 4-980e2-1.1-c1e2-0-0
Degree $4$
Conductor $960400$
Sign $1$
Analytic cond. $61.2359$
Root an. cond. $2.79738$
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  − 3·3-s + 5-s + 3·9-s + 5·11-s − 6·13-s − 3·15-s + 17-s − 6·19-s − 6·23-s − 18·29-s + 4·31-s − 15·33-s − 2·37-s + 18·39-s − 8·41-s + 20·43-s + 3·45-s + 47-s − 3·51-s − 4·53-s + 5·55-s + 18·57-s + 8·59-s + 8·61-s − 6·65-s − 12·67-s + 18·69-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 9-s + 1.50·11-s − 1.66·13-s − 0.774·15-s + 0.242·17-s − 1.37·19-s − 1.25·23-s − 3.34·29-s + 0.718·31-s − 2.61·33-s − 0.328·37-s + 2.88·39-s − 1.24·41-s + 3.04·43-s + 0.447·45-s + 0.145·47-s − 0.420·51-s − 0.549·53-s + 0.674·55-s + 2.38·57-s + 1.04·59-s + 1.02·61-s − 0.744·65-s − 1.46·67-s + 2.16·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(960400\)    =    \(2^{4} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(61.2359\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3456594910\)
\(L(\frac12)\) \(\approx\) \(0.3456594910\)
\(L(\frac{3}{2})\) 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)$
bad2 \( 1 \)
5$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
good3$C_2$ \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \)
11$C_2^2$ \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 8 T + 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 4 T - 73 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
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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

−10.10029021743968538023671421925, −9.934430994249639686033959697071, −9.374713437772091599844849458417, −9.267628747341843134584118262387, −8.563787333573754041189024056033, −8.140181850176752005969610980992, −7.44993758185122295633808126641, −7.13796872600580535725060506007, −6.79469758201643593411032712868, −6.14958816115782488649017942668, −5.91614061787608399566901158135, −5.58222151954653710203048705420, −5.19033671783751511397496297411, −4.52325463981775265974686942944, −3.99935389507306078874350177370, −3.81359052049937885899183199778, −2.64709451011125062993507827783, −2.09494057477336641200053054371, −1.48724060963417655401236893420, −0.30127134837684575171694354925, 0.30127134837684575171694354925, 1.48724060963417655401236893420, 2.09494057477336641200053054371, 2.64709451011125062993507827783, 3.81359052049937885899183199778, 3.99935389507306078874350177370, 4.52325463981775265974686942944, 5.19033671783751511397496297411, 5.58222151954653710203048705420, 5.91614061787608399566901158135, 6.14958816115782488649017942668, 6.79469758201643593411032712868, 7.13796872600580535725060506007, 7.44993758185122295633808126641, 8.140181850176752005969610980992, 8.563787333573754041189024056033, 9.267628747341843134584118262387, 9.374713437772091599844849458417, 9.934430994249639686033959697071, 10.10029021743968538023671421925

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