Properties

Label 4-363e2-1.1-c7e2-0-0
Degree $4$
Conductor $131769$
Sign $1$
Analytic cond. $12858.5$
Root an. cond. $10.6487$
Motivic weight $7$
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  + 19·2-s + 54·3-s + 59·4-s − 34·5-s + 1.02e3·6-s + 166·7-s − 2.18e3·8-s + 2.18e3·9-s − 646·10-s + 3.18e3·12-s + 1.26e4·13-s + 3.15e3·14-s − 1.83e3·15-s − 2.95e4·16-s + 6.23e4·17-s + 4.15e4·18-s + 3.79e4·19-s − 2.00e3·20-s + 8.96e3·21-s − 9.06e4·23-s − 1.17e5·24-s − 9.14e4·25-s + 2.40e5·26-s + 7.87e4·27-s + 9.79e3·28-s − 1.39e4·29-s − 3.48e4·30-s + ⋯
L(s)  = 1  + 1.67·2-s + 1.15·3-s + 0.460·4-s − 0.121·5-s + 1.93·6-s + 0.182·7-s − 1.50·8-s + 9-s − 0.204·10-s + 0.532·12-s + 1.59·13-s + 0.307·14-s − 0.140·15-s − 1.80·16-s + 3.07·17-s + 1.67·18-s + 1.27·19-s − 0.0560·20-s + 0.211·21-s − 1.55·23-s − 1.74·24-s − 1.17·25-s + 2.68·26-s + 0.769·27-s + 0.0843·28-s − 0.106·29-s − 0.235·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131769 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(131769\)    =    \(3^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(12858.5\)
Root analytic conductor: \(10.6487\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 131769,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(18.03740582\)
\(L(\frac12)\) \(\approx\) \(18.03740582\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 - p^{3} T )^{2} \)
11 \( 1 \)
good2$D_{4}$ \( 1 - 19 T + 151 p T^{2} - 19 p^{7} T^{3} + p^{14} T^{4} \)
5$D_{4}$ \( 1 + 34 T + 92642 T^{2} + 34 p^{7} T^{3} + p^{14} T^{4} \)
7$D_{4}$ \( 1 - 166 T + 604542 T^{2} - 166 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 - 12670 T + 131517642 T^{2} - 12670 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 62344 T + 1748474222 T^{2} - 62344 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 37980 T + 2116242326 T^{2} - 37980 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 + 90686 T + 13407542 p^{2} T^{2} + 90686 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 13992 T - 4336731434 T^{2} + 13992 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 - 245000 T + 56312134590 T^{2} - 245000 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 - 327852 T + 183339535550 T^{2} - 327852 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 - 275932 T + 66680638310 T^{2} - 275932 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 - 244104 T + 335871625670 T^{2} - 244104 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 - 536926 T + 837168473222 T^{2} - 536926 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 1821882 T + 2746994724802 T^{2} + 1821882 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 2502028 T + 4952639534534 T^{2} - 2502028 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 - 2191098 T + 5681315972690 T^{2} - 2191098 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 1674784 T + 3291529437702 T^{2} - 1674784 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 + 368310 T + 18165736320454 T^{2} + 368310 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 - 3336604 T + 16427959744566 T^{2} - 3336604 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 1682618 T + 16894191271566 T^{2} - 1682618 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 - 8376504 T + 62394156052870 T^{2} - 8376504 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 9027204 T + 72708018275350 T^{2} - 9027204 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 + 16703552 T + 231348397156350 T^{2} + 16703552 p^{7} T^{3} + p^{14} T^{4} \)
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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

−10.26917419384440731727832094794, −9.935732606735947169533601764981, −9.406421447232998287769796592230, −9.372082595965048328396455039265, −8.221521153436092804094641476956, −8.170156802012258876694948270101, −7.906139359033015612409791135086, −7.26682870610285780208554409382, −6.25169314653622172526571229524, −6.07000117769029323381244882298, −5.40895324190330384920110695582, −5.23376602759423292729189002906, −4.26118214519703633654505119354, −4.02416566216348484785921991057, −3.43450968493403648223238468895, −3.35383394715594240317339924681, −2.66128252885637057277631845441, −1.83190981019525032527058786261, −0.992423085763486885647394327912, −0.76168788967553638584873809393, 0.76168788967553638584873809393, 0.992423085763486885647394327912, 1.83190981019525032527058786261, 2.66128252885637057277631845441, 3.35383394715594240317339924681, 3.43450968493403648223238468895, 4.02416566216348484785921991057, 4.26118214519703633654505119354, 5.23376602759423292729189002906, 5.40895324190330384920110695582, 6.07000117769029323381244882298, 6.25169314653622172526571229524, 7.26682870610285780208554409382, 7.906139359033015612409791135086, 8.170156802012258876694948270101, 8.221521153436092804094641476956, 9.372082595965048328396455039265, 9.406421447232998287769796592230, 9.935732606735947169533601764981, 10.26917419384440731727832094794

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