Properties

Label 4-5040e2-1.1-c1e2-0-20
Degree $4$
Conductor $25401600$
Sign $1$
Analytic cond. $1619.62$
Root an. cond. $6.34386$
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  − 2·5-s + 2·7-s + 11-s + 5·13-s + 5·17-s + 6·19-s − 2·23-s + 3·25-s − 29-s − 4·35-s + 12·37-s − 2·41-s − 10·43-s − 5·47-s + 3·49-s + 2·53-s − 2·55-s − 8·59-s + 6·61-s − 10·65-s − 4·67-s + 16·71-s − 8·73-s + 2·77-s + 9·79-s + 8·83-s − 10·85-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.755·7-s + 0.301·11-s + 1.38·13-s + 1.21·17-s + 1.37·19-s − 0.417·23-s + 3/5·25-s − 0.185·29-s − 0.676·35-s + 1.97·37-s − 0.312·41-s − 1.52·43-s − 0.729·47-s + 3/7·49-s + 0.274·53-s − 0.269·55-s − 1.04·59-s + 0.768·61-s − 1.24·65-s − 0.488·67-s + 1.89·71-s − 0.936·73-s + 0.227·77-s + 1.01·79-s + 0.878·83-s − 1.08·85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25401600\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1619.62\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5040} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25401600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.975965693\)
\(L(\frac12)\) \(\approx\) \(3.975965693\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$D_{4}$ \( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 28 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + T + 20 T^{2} + p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 2 T + 66 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 94 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 5 T + 62 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 6 T - 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 9 T + 174 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 + 6 T + 170 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 9 T + 108 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−8.142714943473368684993814615183, −8.122446512639030530693895560621, −7.76860583892026575047000319187, −7.51371198676106616266290266055, −7.02971819149202655701404643524, −6.65645291240195735906747375761, −6.18832360619498556644098522520, −5.96324927266163111427330138227, −5.35335882065032152235231234975, −5.26176791022635776098895890764, −4.65007125380759800348773691451, −4.37791204752379988825121286000, −3.80471134928957207647720778826, −3.61858962062271477339498685643, −3.00601330755127367252021836079, −3.00305092159029934898427042191, −1.85410117085905040543520141482, −1.73494510177470132307942603571, −0.867016930643069539805314838033, −0.74422336636659750129507060410, 0.74422336636659750129507060410, 0.867016930643069539805314838033, 1.73494510177470132307942603571, 1.85410117085905040543520141482, 3.00305092159029934898427042191, 3.00601330755127367252021836079, 3.61858962062271477339498685643, 3.80471134928957207647720778826, 4.37791204752379988825121286000, 4.65007125380759800348773691451, 5.26176791022635776098895890764, 5.35335882065032152235231234975, 5.96324927266163111427330138227, 6.18832360619498556644098522520, 6.65645291240195735906747375761, 7.02971819149202655701404643524, 7.51371198676106616266290266055, 7.76860583892026575047000319187, 8.122446512639030530693895560621, 8.142714943473368684993814615183

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