Properties

Label 4-69e2-1.1-c3e2-0-0
Degree $4$
Conductor $4761$
Sign $1$
Analytic cond. $16.5741$
Root an. cond. $2.01770$
Motivic weight $3$
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  + 4·3-s − 7·4-s − 11·9-s − 28·12-s + 148·13-s − 15·16-s − 250·25-s − 152·27-s + 688·31-s + 77·36-s + 592·39-s − 60·48-s + 686·49-s − 1.03e3·52-s + 553·64-s + 2.45e3·73-s − 1.00e3·75-s − 311·81-s + 2.75e3·93-s + 1.75e3·100-s + 1.06e3·108-s − 1.62e3·117-s − 2.66e3·121-s − 4.81e3·124-s + 127-s + 131-s + 137-s + ⋯
L(s)  = 1  + 0.769·3-s − 7/8·4-s − 0.407·9-s − 0.673·12-s + 3.15·13-s − 0.234·16-s − 2·25-s − 1.08·27-s + 3.98·31-s + 0.356·36-s + 2.43·39-s − 0.180·48-s + 2·49-s − 2.76·52-s + 1.08·64-s + 3.93·73-s − 1.53·75-s − 0.426·81-s + 3.06·93-s + 7/4·100-s + 0.947·108-s − 1.28·117-s − 2·121-s − 3.48·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4761\)    =    \(3^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(16.5741\)
Root analytic conductor: \(2.01770\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{69} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4761,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.976665600\)
\(L(\frac12)\) \(\approx\) \(1.976665600\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 - 4 T + p^{3} T^{2} \)
23$C_2$ \( 1 + p^{3} T^{2} \)
good2$C_2$ \( ( 1 - 3 T + p^{3} T^{2} )( 1 + 3 T + p^{3} T^{2} ) \)
5$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
7$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
11$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 74 T + p^{3} T^{2} )^{2} \)
17$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
19$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 - 282 T + p^{3} T^{2} )( 1 + 282 T + p^{3} T^{2} ) \)
31$C_2$ \( ( 1 - 344 T + p^{3} T^{2} )^{2} \)
37$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
41$C_2$ \( ( 1 - 426 T + p^{3} T^{2} )( 1 + 426 T + p^{3} T^{2} ) \)
43$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
47$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )( 1 + 48 T + p^{3} T^{2} ) \)
53$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
59$C_2$ \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \)
61$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
67$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 1176 T + p^{3} T^{2} )( 1 + 1176 T + p^{3} T^{2} ) \)
73$C_2$ \( ( 1 - 1226 T + p^{3} T^{2} )^{2} \)
79$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
89$C_2$ \( ( 1 + p^{3} T^{2} )^{2} \)
97$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
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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

−15.27421098183021062462505979863, −14.00130746800366898984595644535, −13.45471690948708818404110321292, −13.44994154409642373551389140765, −12.37440759397331378666586814780, −11.56317134080674220664646831262, −11.35819148611223138738775427391, −10.48819360561422095006563385658, −9.900023497948015880628023172706, −9.134582039500908511314170686275, −8.748361864258654657959388936913, −8.134491939867327801130737487398, −7.967895682497640834627472920994, −6.38010374506982623737782445471, −6.23143845428292983270152484031, −5.22147457768017450620088117595, −3.98297866686422702784859810415, −3.80305279376031239900559755841, −2.53801662311772924551648335151, −0.996139940795100827595483904932, 0.996139940795100827595483904932, 2.53801662311772924551648335151, 3.80305279376031239900559755841, 3.98297866686422702784859810415, 5.22147457768017450620088117595, 6.23143845428292983270152484031, 6.38010374506982623737782445471, 7.967895682497640834627472920994, 8.134491939867327801130737487398, 8.748361864258654657959388936913, 9.134582039500908511314170686275, 9.900023497948015880628023172706, 10.48819360561422095006563385658, 11.35819148611223138738775427391, 11.56317134080674220664646831262, 12.37440759397331378666586814780, 13.44994154409642373551389140765, 13.45471690948708818404110321292, 14.00130746800366898984595644535, 15.27421098183021062462505979863

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