Properties

Label 4-560e2-1.1-c1e2-0-14
Degree $4$
Conductor $313600$
Sign $1$
Analytic cond. $19.9954$
Root an. cond. $2.11462$
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 + 5·9-s + 6·11-s + 11·25-s + 10·29-s − 4·31-s + 4·41-s − 20·45-s − 49-s − 24·55-s + 20·59-s − 16·61-s + 16·71-s − 10·79-s + 16·81-s + 30·99-s + 24·101-s − 10·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 1.78·5-s + 5/3·9-s + 1.80·11-s + 11/5·25-s + 1.85·29-s − 0.718·31-s + 0.624·41-s − 2.98·45-s − 1/7·49-s − 3.23·55-s + 2.60·59-s − 2.04·61-s + 1.89·71-s − 1.12·79-s + 16/9·81-s + 3.01·99-s + 2.38·101-s − 0.957·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(313600\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.9954\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{560} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 313600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722027016\)
\(L(\frac12)\) \(\approx\) \(1.722027016\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−10.91070189481907922956927318156, −10.71009106781184143203315927908, −10.03091677087469056227918960742, −9.766656387781431151823788433392, −9.129171517005568276859253511243, −8.763191837037744281387501238879, −8.375848115054907090016583855751, −7.70808153712614068374976575634, −7.45311496359253204808358138568, −7.00666108758363224630411901529, −6.46838322567371333364017657592, −6.35389253844449468960839880985, −5.17906672504472111908101207858, −4.74636824709949268335118783326, −4.13719132333362772741355965576, −3.94513573682298272413934743563, −3.50828238529535548574726346732, −2.61115291801782959244242904270, −1.48207310990774027324936125367, −0.873193878614773183523832811103, 0.873193878614773183523832811103, 1.48207310990774027324936125367, 2.61115291801782959244242904270, 3.50828238529535548574726346732, 3.94513573682298272413934743563, 4.13719132333362772741355965576, 4.74636824709949268335118783326, 5.17906672504472111908101207858, 6.35389253844449468960839880985, 6.46838322567371333364017657592, 7.00666108758363224630411901529, 7.45311496359253204808358138568, 7.70808153712614068374976575634, 8.375848115054907090016583855751, 8.763191837037744281387501238879, 9.129171517005568276859253511243, 9.766656387781431151823788433392, 10.03091677087469056227918960742, 10.71009106781184143203315927908, 10.91070189481907922956927318156

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