Properties

Label 4-69e2-1.1-c5e2-0-0
Degree $4$
Conductor $4761$
Sign $1$
Analytic cond. $122.467$
Root an. cond. $3.32663$
Motivic weight $5$
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  − 18·3-s + 52·4-s + 94·5-s − 118·7-s + 243·9-s + 320·11-s − 936·12-s − 288·13-s − 1.69e3·15-s + 1.68e3·16-s − 1.81e3·17-s + 730·19-s + 4.88e3·20-s + 2.12e3·21-s + 1.05e3·23-s + 406·25-s − 2.91e3·27-s − 6.13e3·28-s + 8.20e3·29-s + 1.77e3·31-s − 5.76e3·33-s − 1.10e4·35-s + 1.26e4·36-s − 2.31e4·37-s + 5.18e3·39-s + 5.51e3·41-s + 1.03e4·43-s + ⋯
L(s)  = 1  − 1.15·3-s + 13/8·4-s + 1.68·5-s − 0.910·7-s + 9-s + 0.797·11-s − 1.87·12-s − 0.472·13-s − 1.94·15-s + 1.64·16-s − 1.51·17-s + 0.463·19-s + 2.73·20-s + 1.05·21-s + 0.417·23-s + 0.129·25-s − 0.769·27-s − 1.47·28-s + 1.81·29-s + 0.331·31-s − 0.920·33-s − 1.53·35-s + 13/8·36-s − 2.77·37-s + 0.545·39-s + 0.512·41-s + 0.851·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4761\)    =    \(3^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(122.467\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{69} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4761,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(3.084529118\)
\(L(\frac12)\) \(\approx\) \(3.084529118\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{2} \)
23$C_1$ \( ( 1 - p^{2} T )^{2} \)
good2$C_2^2$ \( 1 - 13 p^{2} T^{2} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 94 T + 1686 p T^{2} - 94 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 + 118 T + 4798 p T^{2} + 118 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 320 T + 9446 T^{2} - 320 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 288 T + 604518 T^{2} + 288 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 1810 T + 3653838 T^{2} + 1810 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 - 730 T + 2280514 T^{2} - 730 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 8208 T + 55813190 T^{2} - 8208 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 1772 T - 8430386 T^{2} - 1772 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 23112 T + 270998406 T^{2} + 23112 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 5516 T + 184729830 T^{2} - 5516 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 10322 T + 114379026 T^{2} - 10322 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 42952 T + 881422574 T^{2} - 42952 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 25350 T + 443489942 T^{2} + 25350 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 18344 T + 1506490326 T^{2} - 18344 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 37224 T + 1584042582 T^{2} - 37224 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 7482 T + 31324422 p T^{2} + 7482 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 126848 T + 7504629902 T^{2} - 126848 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 137660 T + 8741816710 T^{2} - 137660 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 62286 T + 7060563458 T^{2} - 62286 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 83120 T + 6016756982 T^{2} - 83120 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 - 69770 T + 12229395942 T^{2} - 69770 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 170104 T + 21347426014 T^{2} + 170104 p^{5} T^{3} + p^{10} T^{4} \)
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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

−13.73060252542277886036244053644, −13.69217455460158908780875074479, −12.68165734366132744672456196108, −12.16031416838799569990171706179, −12.02896651631337138325935318147, −11.01103954147332120567672158061, −10.80754337700588725386412592380, −10.23181548528104791341494476831, −9.510942592783838027409499138020, −9.299326725517822566666347237099, −8.056400359726645824371823969231, −6.85340485695555742582358487104, −6.71296593317586937400900470431, −6.39984799630137542812634852709, −5.64430241437460740915080945928, −5.03856357092265589808621599610, −3.70839814603054667537061206146, −2.43766725803651935698219596128, −1.98223703096923959271753211523, −0.842382676172294653687089750060, 0.842382676172294653687089750060, 1.98223703096923959271753211523, 2.43766725803651935698219596128, 3.70839814603054667537061206146, 5.03856357092265589808621599610, 5.64430241437460740915080945928, 6.39984799630137542812634852709, 6.71296593317586937400900470431, 6.85340485695555742582358487104, 8.056400359726645824371823969231, 9.299326725517822566666347237099, 9.510942592783838027409499138020, 10.23181548528104791341494476831, 10.80754337700588725386412592380, 11.01103954147332120567672158061, 12.02896651631337138325935318147, 12.16031416838799569990171706179, 12.68165734366132744672456196108, 13.69217455460158908780875074479, 13.73060252542277886036244053644

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