Properties

Label 4-3120e2-1.1-c1e2-0-3
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
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  + 4·5-s − 9-s − 4·11-s − 8·19-s + 11·25-s − 8·29-s − 16·31-s − 12·41-s − 4·45-s + 10·49-s − 16·55-s + 20·59-s − 28·61-s + 8·71-s − 16·79-s + 81-s − 12·89-s − 32·95-s + 4·99-s + 8·101-s + 36·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 1.20·11-s − 1.83·19-s + 11/5·25-s − 1.48·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s − 2.15·55-s + 2.60·59-s − 3.58·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s − 3.28·95-s + 0.402·99-s + 0.796·101-s + 3.44·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3120} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.040961068\)
\(L(\frac12)\) \(\approx\) \(1.040961068\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−9.133268811439705425744653861659, −8.603798841262593642195515003299, −8.344736712617183097635433907133, −7.57750170309230984537305192346, −7.45040171591637541988342342090, −6.89687004795153531334743827776, −6.71509816029360762341239124059, −5.94947025571532300547259944484, −5.86624795008074368619694933757, −5.55896621300738060861822604438, −5.21686885616493223095949956927, −4.72234377214669647550459857644, −4.31890707998252421772462088689, −3.51500931558475101007930422432, −3.43088109584937124432054037674, −2.53329523386891776614274580468, −2.32225797222956469181091582995, −1.83546881909555737139101018036, −1.52987831900190524830934529347, −0.28491494573742795891265509846, 0.28491494573742795891265509846, 1.52987831900190524830934529347, 1.83546881909555737139101018036, 2.32225797222956469181091582995, 2.53329523386891776614274580468, 3.43088109584937124432054037674, 3.51500931558475101007930422432, 4.31890707998252421772462088689, 4.72234377214669647550459857644, 5.21686885616493223095949956927, 5.55896621300738060861822604438, 5.86624795008074368619694933757, 5.94947025571532300547259944484, 6.71509816029360762341239124059, 6.89687004795153531334743827776, 7.45040171591637541988342342090, 7.57750170309230984537305192346, 8.344736712617183097635433907133, 8.603798841262593642195515003299, 9.133268811439705425744653861659

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