Properties

Label 4-4680e2-1.1-c1e2-0-7
Degree $4$
Conductor $21902400$
Sign $1$
Analytic cond. $1396.51$
Root an. cond. $6.11309$
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  − 2·5-s − 7-s + 7·11-s + 2·13-s + 17-s − 2·19-s + 7·23-s + 3·25-s + 4·29-s − 6·31-s + 2·35-s − 37-s + 7·41-s − 4·43-s − 6·47-s − 9·49-s − 3·53-s − 14·55-s + 16·59-s − 9·61-s − 4·65-s − 12·67-s + 3·71-s + 16·73-s − 7·77-s + 3·79-s + 16·83-s + ⋯
L(s)  = 1  − 0.894·5-s − 0.377·7-s + 2.11·11-s + 0.554·13-s + 0.242·17-s − 0.458·19-s + 1.45·23-s + 3/5·25-s + 0.742·29-s − 1.07·31-s + 0.338·35-s − 0.164·37-s + 1.09·41-s − 0.609·43-s − 0.875·47-s − 9/7·49-s − 0.412·53-s − 1.88·55-s + 2.08·59-s − 1.15·61-s − 0.496·65-s − 1.46·67-s + 0.356·71-s + 1.87·73-s − 0.797·77-s + 0.337·79-s + 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21902400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(21902400\)    =    \(2^{6} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1396.51\)
Root analytic conductor: \(6.11309\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4680} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 21902400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.824984513\)
\(L(\frac12)\) \(\approx\) \(2.824984513\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 - T )^{2} \)
good7$D_{4}$ \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 7 T + 54 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 + 6 T + 54 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + T - 32 T^{2} + p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 7 T + 56 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 3 T + 54 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 17 T + 212 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 15 T + 144 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
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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

−8.495051124554728999223797716528, −8.156172213138674823885627091360, −7.69738746027565250462421092569, −7.50169714488355958654239591087, −6.86665299028155737679554142139, −6.63786260201291640589841428263, −6.44612833185801654666308918877, −6.21385289254595971035196991562, −5.40042644848309558008843104383, −5.26350848774820290129026635031, −4.63202992754454959423379687598, −4.35906421335095256086661752033, −3.88273734661303330469144670286, −3.60558390242330559636734911977, −3.20266951605659161800882819320, −2.91972495491246424361349917833, −2.02124778414735920230086391383, −1.63928801998691322522002884310, −0.985773035544421462851908715762, −0.56947844604877119435785832240, 0.56947844604877119435785832240, 0.985773035544421462851908715762, 1.63928801998691322522002884310, 2.02124778414735920230086391383, 2.91972495491246424361349917833, 3.20266951605659161800882819320, 3.60558390242330559636734911977, 3.88273734661303330469144670286, 4.35906421335095256086661752033, 4.63202992754454959423379687598, 5.26350848774820290129026635031, 5.40042644848309558008843104383, 6.21385289254595971035196991562, 6.44612833185801654666308918877, 6.63786260201291640589841428263, 6.86665299028155737679554142139, 7.50169714488355958654239591087, 7.69738746027565250462421092569, 8.156172213138674823885627091360, 8.495051124554728999223797716528

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