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Properties

Label 4-69e2-1.1-c1e2-0-0
Degree $4$
Conductor $4761$
Sign $1$
Analytic cond. $0.303565$
Root an. cond. $0.742272$
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  − 2·3-s + 4-s − 2·5-s + 2·7-s + 3·9-s + 8·11-s − 2·12-s + 4·15-s − 3·16-s − 10·17-s + 10·19-s − 2·20-s − 4·21-s + 2·23-s − 2·25-s − 4·27-s + 2·28-s − 4·31-s − 16·33-s − 4·35-s + 3·36-s − 4·41-s + 2·43-s + 8·44-s − 6·45-s − 8·47-s + 6·48-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 9-s + 2.41·11-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 2.42·17-s + 2.29·19-s − 0.447·20-s − 0.872·21-s + 0.417·23-s − 2/5·25-s − 0.769·27-s + 0.377·28-s − 0.718·31-s − 2.78·33-s − 0.676·35-s + 1/2·36-s − 0.624·41-s + 0.304·43-s + 1.20·44-s − 0.894·45-s − 1.16·47-s + 0.866·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4761\)    =    \(3^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(0.303565\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4761,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6600896958\)
\(L(\frac12)\) \(\approx\) \(0.6600896958\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
5$D_{4}$ \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
41$C_4$ \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−14.95715533007426163850111930637, −14.76183905833454068029728432404, −13.81899745768686567473769273524, −13.51466322905824576478557971092, −12.59965045147381510169012967478, −11.90798729347742989362181053457, −11.53274189797156810805179065395, −11.28536519904873953585469610543, −11.17055123636336968773405136761, −9.998840008183561101707583288020, −9.209997901439462058565944429156, −8.913064862793878452226070374158, −7.85443001137786965466864754159, −7.07227181138753725370161539746, −6.76275085319516488596762758674, −6.12291869017769029251338113344, −4.98567104290795192227696532437, −4.41931729379675550092161587321, −3.61218196608973072274220972099, −1.65001680377957710884516373019, 1.65001680377957710884516373019, 3.61218196608973072274220972099, 4.41931729379675550092161587321, 4.98567104290795192227696532437, 6.12291869017769029251338113344, 6.76275085319516488596762758674, 7.07227181138753725370161539746, 7.85443001137786965466864754159, 8.913064862793878452226070374158, 9.209997901439462058565944429156, 9.998840008183561101707583288020, 11.17055123636336968773405136761, 11.28536519904873953585469610543, 11.53274189797156810805179065395, 11.90798729347742989362181053457, 12.59965045147381510169012967478, 13.51466322905824576478557971092, 13.81899745768686567473769273524, 14.76183905833454068029728432404, 14.95715533007426163850111930637

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