Properties

Label 4-7360e2-1.1-c1e2-0-1
Degree $4$
Conductor $54169600$
Sign $1$
Analytic cond. $3453.90$
Root an. cond. $7.66615$
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-s − 2·5-s − 7-s − 4·9-s + 11-s + 3·13-s − 2·15-s + 17-s − 3·19-s − 21-s − 2·23-s + 3·25-s − 6·27-s + 14·29-s − 7·31-s + 33-s + 2·35-s − 4·37-s + 3·39-s − 9·41-s + 8·45-s − 6·47-s − 12·49-s + 51-s + 8·53-s − 2·55-s − 3·57-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s − 4/3·9-s + 0.301·11-s + 0.832·13-s − 0.516·15-s + 0.242·17-s − 0.688·19-s − 0.218·21-s − 0.417·23-s + 3/5·25-s − 1.15·27-s + 2.59·29-s − 1.25·31-s + 0.174·33-s + 0.338·35-s − 0.657·37-s + 0.480·39-s − 1.40·41-s + 1.19·45-s − 0.875·47-s − 1.71·49-s + 0.140·51-s + 1.09·53-s − 0.269·55-s − 0.397·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54169600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54169600\)    =    \(2^{12} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3453.90\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 54169600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.568634596\)
\(L(\frac12)\) \(\approx\) \(1.568634596\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 13 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T - 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 14 T + 102 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 7 T + 43 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 3 T + 63 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 3 T + 113 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 12 T + 174 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
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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

−8.092545752927417875603902820627, −7.971393399209295323623329893488, −7.47262979578302811213487256580, −6.96631091565246180415154073091, −6.59111876934830152893849690710, −6.50804598848037437464080577868, −6.06531838266606260376281125285, −5.65729697233838068584139532940, −5.23925111121201693835090570713, −4.86532878554780819358598248907, −4.50908342568908667971514163295, −4.01565402326033819922029811916, −3.51964046435717932940837699915, −3.46183733204483374424908397305, −2.92938579063873202036819007700, −2.77451841214326183808744900619, −1.84583387825285221848584750045, −1.83546086304153058285364351019, −0.837588161432488574259431249259, −0.36097499674669535031112982929, 0.36097499674669535031112982929, 0.837588161432488574259431249259, 1.83546086304153058285364351019, 1.84583387825285221848584750045, 2.77451841214326183808744900619, 2.92938579063873202036819007700, 3.46183733204483374424908397305, 3.51964046435717932940837699915, 4.01565402326033819922029811916, 4.50908342568908667971514163295, 4.86532878554780819358598248907, 5.23925111121201693835090570713, 5.65729697233838068584139532940, 6.06531838266606260376281125285, 6.50804598848037437464080577868, 6.59111876934830152893849690710, 6.96631091565246180415154073091, 7.47262979578302811213487256580, 7.971393399209295323623329893488, 8.092545752927417875603902820627

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