Properties

Label 4-9702e2-1.1-c1e2-0-5
Degree $4$
Conductor $94128804$
Sign $1$
Analytic cond. $6001.73$
Root an. cond. $8.80175$
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 + 3·4-s + 4·5-s − 4·8-s − 8·10-s − 2·11-s − 2·13-s + 5·16-s + 4·17-s − 4·19-s + 12·20-s + 4·22-s + 4·23-s + 4·25-s + 4·26-s + 6·29-s − 8·31-s − 6·32-s − 8·34-s − 16·37-s + 8·38-s − 16·40-s + 8·41-s − 6·44-s − 8·46-s + 4·47-s − 8·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 1.78·5-s − 1.41·8-s − 2.52·10-s − 0.603·11-s − 0.554·13-s + 5/4·16-s + 0.970·17-s − 0.917·19-s + 2.68·20-s + 0.852·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s + 1.11·29-s − 1.43·31-s − 1.06·32-s − 1.37·34-s − 2.63·37-s + 1.29·38-s − 2.52·40-s + 1.24·41-s − 0.904·44-s − 1.17·46-s + 0.583·47-s − 1.13·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(94128804\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(6001.73\)
Root analytic conductor: \(8.80175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{9702} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 94128804,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.160539152\)
\(L(\frac12)\) \(\approx\) \(2.160539152\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_4$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 32 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 + 16 T + 136 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 8 T + 96 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 18 T + 195 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 14 T + 165 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 108 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 16 T + 208 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 18 T + 221 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 4 T - 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 187 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.76802327616644165772520562307, −7.51183389471990926566209118894, −7.19576411792399136382220659439, −6.94261568533896605644906872199, −6.51882453803484115696885308398, −6.36626956914172911989573151872, −5.60378833607157166717260971034, −5.58394062210325095176295216647, −5.37949360316654319972706474199, −5.09245245511981730746161516684, −4.18720382324560450632702719503, −4.13420906498941461097151240478, −3.25167360235717898908621419292, −3.16548352747667186226729363080, −2.40326634182346310482063540030, −2.36802872553764179731784778409, −1.80070410638963361008062689809, −1.64739423734673758886223729563, −0.842803186622131713617098643063, −0.49362482980919079971154421053, 0.49362482980919079971154421053, 0.842803186622131713617098643063, 1.64739423734673758886223729563, 1.80070410638963361008062689809, 2.36802872553764179731784778409, 2.40326634182346310482063540030, 3.16548352747667186226729363080, 3.25167360235717898908621419292, 4.13420906498941461097151240478, 4.18720382324560450632702719503, 5.09245245511981730746161516684, 5.37949360316654319972706474199, 5.58394062210325095176295216647, 5.60378833607157166717260971034, 6.36626956914172911989573151872, 6.51882453803484115696885308398, 6.94261568533896605644906872199, 7.19576411792399136382220659439, 7.51183389471990926566209118894, 7.76802327616644165772520562307

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