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Properties

Label 4-93e2-1.1-c1e2-0-2
Degree $4$
Conductor $8649$
Sign $1$
Analytic cond. $0.551467$
Root an. cond. $0.861747$
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  + 3·3-s + 4-s − 3·5-s + 7-s + 6·9-s − 3·11-s + 3·12-s − 9·13-s − 9·15-s − 3·16-s + 3·17-s + 5·19-s − 3·20-s + 3·21-s + 25-s + 9·27-s + 28-s + 12·29-s + 4·31-s − 9·33-s − 3·35-s + 6·36-s − 9·37-s − 27·39-s − 15·41-s + 3·43-s − 3·44-s + ⋯
L(s)  = 1  + 1.73·3-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 2·9-s − 0.904·11-s + 0.866·12-s − 2.49·13-s − 2.32·15-s − 3/4·16-s + 0.727·17-s + 1.14·19-s − 0.670·20-s + 0.654·21-s + 1/5·25-s + 1.73·27-s + 0.188·28-s + 2.22·29-s + 0.718·31-s − 1.56·33-s − 0.507·35-s + 36-s − 1.47·37-s − 4.32·39-s − 2.34·41-s + 0.457·43-s − 0.452·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8649\)    =    \(3^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(0.551467\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8649,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.385274970\)
\(L(\frac12)\) \(\approx\) \(1.385274970\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 - p T + p T^{2} \)
31$C_2$ \( 1 - 4 T + p T^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$C_2^2$ \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
7$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 15 T + 134 T^{2} + 15 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
71$C_2^2$ \( 1 - 3 T + 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 - 15 T + 148 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 15 T + 154 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−14.14318749439639379518622306868, −13.90668270401850848298854495461, −13.63893957038420301761520119000, −12.39762713021177518453692421934, −12.23879883907690774104344662655, −11.98826275339044362396092324811, −11.15740156352528469782352296596, −10.29356113648857766182774869242, −9.963222189697034094609711554900, −9.439863946897380853111640817778, −8.463562627703474581601037806054, −8.221441931081434814022982298753, −7.46602996119205449573572430003, −7.41229624837991592986350460284, −6.70900194422600263127347407529, −4.93567012356677985048798995981, −4.88221174620953294454735658044, −3.65914927120740207585272776376, −2.91568675286502753385018921932, −2.26841495158615668103322489490, 2.26841495158615668103322489490, 2.91568675286502753385018921932, 3.65914927120740207585272776376, 4.88221174620953294454735658044, 4.93567012356677985048798995981, 6.70900194422600263127347407529, 7.41229624837991592986350460284, 7.46602996119205449573572430003, 8.221441931081434814022982298753, 8.463562627703474581601037806054, 9.439863946897380853111640817778, 9.963222189697034094609711554900, 10.29356113648857766182774869242, 11.15740156352528469782352296596, 11.98826275339044362396092324811, 12.23879883907690774104344662655, 12.39762713021177518453692421934, 13.63893957038420301761520119000, 13.90668270401850848298854495461, 14.14318749439639379518622306868

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