Properties

Label 4-60e3-1.1-c1e2-0-5
Degree $4$
Conductor $216000$
Sign $1$
Analytic cond. $13.7723$
Root an. cond. $1.92642$
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-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s + 12-s + 4·13-s + 15-s − 16-s + 4·17-s − 18-s + 8·19-s + 20-s − 3·24-s + 25-s − 4·26-s − 27-s + 4·29-s − 30-s − 5·32-s − 4·34-s − 36-s + 20·37-s − 8·38-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 1.10·13-s + 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.612·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s + 0.742·29-s − 0.182·30-s − 0.883·32-s − 0.685·34-s − 1/6·36-s + 3.28·37-s − 1.29·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(216000\)    =    \(2^{6} \cdot 3^{3} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(13.7723\)
Root analytic conductor: \(1.92642\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{216000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 216000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9303941189\)
\(L(\frac12)\) \(\approx\) \(0.9303941189\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 + T \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−9.327820441539330420804608673051, −8.455181921164028207665051683361, −8.074746559817531846840486780032, −7.67749235610857150327119620769, −7.46261028773657489832215471574, −6.56365355473852823699216167747, −6.20681526904542088835730053039, −5.63406482120703248001185629214, −4.87700802399738744801917392626, −4.79492945983632365647604814265, −3.76558570458913747879946495025, −3.55714635244582915245852508954, −2.61544872003966305633910980827, −1.27025970316177507740293768624, −0.870154925974925884967934945032, 0.870154925974925884967934945032, 1.27025970316177507740293768624, 2.61544872003966305633910980827, 3.55714635244582915245852508954, 3.76558570458913747879946495025, 4.79492945983632365647604814265, 4.87700802399738744801917392626, 5.63406482120703248001185629214, 6.20681526904542088835730053039, 6.56365355473852823699216167747, 7.46261028773657489832215471574, 7.67749235610857150327119620769, 8.074746559817531846840486780032, 8.455181921164028207665051683361, 9.327820441539330420804608673051

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