Properties

Label 4-980e2-1.1-c1e2-0-17
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: induced by $\chi_{980} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 960400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.136781374\)
\(L(\frac12)\) \(\approx\) \(4.136781374\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \)
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$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
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

−10.07874805288484003330024534610, −9.314339183736443234315971302491, −9.170036079240335566035548870042, −9.144391342023484309781584764377, −8.780991723512763655898270630825, −7.984498717222100246955147740029, −7.84282940941575329803294748095, −7.39754411231380091219373760824, −7.18652279334543349240355008413, −6.22680407899718166068467139003, −5.85384386671571582830167338039, −5.82795065963223808062270303981, −4.75357227039929445424942766486, −3.99431508679137193633347955316, −3.95417231387729679149553435415, −3.29885871641974753156334332248, −3.23561328476068011249769120887, −2.06012136757807741944124969827, −1.86184011198517421290437146900, −0.893051422165929279927283314065, 0.893051422165929279927283314065, 1.86184011198517421290437146900, 2.06012136757807741944124969827, 3.23561328476068011249769120887, 3.29885871641974753156334332248, 3.95417231387729679149553435415, 3.99431508679137193633347955316, 4.75357227039929445424942766486, 5.82795065963223808062270303981, 5.85384386671571582830167338039, 6.22680407899718166068467139003, 7.18652279334543349240355008413, 7.39754411231380091219373760824, 7.84282940941575329803294748095, 7.984498717222100246955147740029, 8.780991723512763655898270630825, 9.144391342023484309781584764377, 9.170036079240335566035548870042, 9.314339183736443234315971302491, 10.07874805288484003330024534610

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