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Properties

Label 4-48e2-1.1-c27e2-0-2
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $49146.7$
Root an. cond. $14.8892$
Motivic weight $27$
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  + 3.18e6·3-s + 2.91e8·5-s − 1.21e11·7-s + 7.62e12·9-s + 2.31e14·11-s − 1.15e15·13-s + 9.29e14·15-s − 4.61e16·17-s + 2.68e17·19-s − 3.87e17·21-s − 1.29e18·23-s + 3.00e18·25-s + 1.62e19·27-s + 6.01e19·29-s − 2.03e20·31-s + 7.39e20·33-s − 3.54e19·35-s + 3.02e21·37-s − 3.68e21·39-s − 1.08e22·41-s + 1.29e21·43-s + 2.22e21·45-s + 9.93e22·47-s − 5.76e22·49-s − 1.47e23·51-s − 3.57e22·53-s + 6.75e22·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.106·5-s − 0.474·7-s + 9-s + 2.02·11-s − 1.05·13-s + 0.123·15-s − 1.12·17-s + 1.46·19-s − 0.547·21-s − 0.537·23-s + 0.402·25-s + 0.769·27-s + 1.08·29-s − 1.49·31-s + 2.33·33-s − 0.0506·35-s + 2.03·37-s − 1.22·39-s − 1.83·41-s + 0.115·43-s + 0.106·45-s + 2.65·47-s − 0.876·49-s − 1.30·51-s − 0.188·53-s + 0.216·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(49146.7\)
Root analytic conductor: \(14.8892\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(\approx\) \(7.609185109\)
\(L(\frac12)\) \(\approx\) \(7.609185109\)
\(L(\frac{29}{2})\) 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)$
bad2 \( 1 \)
3$C_1$ \( ( 1 - p^{13} T )^{2} \)
good5$D_{4}$ \( 1 - 291441036 T - 23327953942428106 p^{3} T^{2} - 291441036 p^{27} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 121646295328 T + \)\(21\!\cdots\!74\)\( p^{3} T^{2} + 121646295328 p^{27} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 21073396524216 p T + \)\(27\!\cdots\!06\)\( p^{3} T^{2} - 21073396524216 p^{28} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 1155759271611764 T + \)\(93\!\cdots\!66\)\( p T^{2} + 1155759271611764 p^{27} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 46125175777027452 T + \)\(75\!\cdots\!66\)\( p T^{2} + 46125175777027452 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 - 14132623934381000 p T + \)\(13\!\cdots\!98\)\( p^{2} T^{2} - 14132623934381000 p^{28} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 56502692435040432 p T + \)\(14\!\cdots\!42\)\( p^{2} T^{2} + 56502692435040432 p^{28} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 - 60178362995785587900 T + \)\(58\!\cdots\!18\)\( T^{2} - 60178362995785587900 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 + \)\(20\!\cdots\!44\)\( T + \)\(46\!\cdots\!06\)\( T^{2} + \)\(20\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 - \)\(30\!\cdots\!88\)\( T + \)\(58\!\cdots\!02\)\( T^{2} - \)\(30\!\cdots\!88\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 + \)\(10\!\cdots\!16\)\( T + \)\(78\!\cdots\!26\)\( T^{2} + \)\(10\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 - \)\(12\!\cdots\!44\)\( T + \)\(19\!\cdots\!98\)\( T^{2} - \)\(12\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 - \)\(99\!\cdots\!32\)\( T + \)\(47\!\cdots\!82\)\( T^{2} - \)\(99\!\cdots\!32\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 + \)\(35\!\cdots\!24\)\( T + \)\(13\!\cdots\!18\)\( T^{2} + \)\(35\!\cdots\!24\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 + \)\(29\!\cdots\!80\)\( T + \)\(12\!\cdots\!38\)\( T^{2} + \)\(29\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(21\!\cdots\!24\)\( T + \)\(33\!\cdots\!86\)\( T^{2} - \)\(21\!\cdots\!24\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 + \)\(13\!\cdots\!08\)\( T + \)\(83\!\cdots\!62\)\( T^{2} + \)\(13\!\cdots\!08\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 - \)\(43\!\cdots\!36\)\( T + \)\(70\!\cdots\!06\)\( T^{2} - \)\(43\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 + \)\(20\!\cdots\!64\)\( T + \)\(49\!\cdots\!18\)\( T^{2} + \)\(20\!\cdots\!64\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 + \)\(56\!\cdots\!00\)\( T + \)\(24\!\cdots\!18\)\( T^{2} + \)\(56\!\cdots\!00\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(46\!\cdots\!04\)\( T + \)\(12\!\cdots\!58\)\( T^{2} - \)\(46\!\cdots\!04\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 + \)\(19\!\cdots\!80\)\( T + \)\(45\!\cdots\!58\)\( T^{2} + \)\(19\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 - \)\(54\!\cdots\!68\)\( T + \)\(94\!\cdots\!82\)\( T^{2} - \)\(54\!\cdots\!68\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−10.94436693500532443005117157072, −10.16615095017602300676912623779, −9.690470117714450293190299982649, −9.330698331600928852650419157591, −8.865676492525466583574453318746, −8.495997507737936948496271897935, −7.48621736765211734997499098291, −7.30796439538500621827271816730, −6.70625691975758057482531836782, −6.18841066958626046258365738347, −5.50476369418244116681712293780, −4.59647152603746605198860374674, −4.28148016975867483145426968632, −3.75246637682478132245336427694, −2.96148040917870694438810242911, −2.88897734265127925217414309474, −1.81038388398900894849820668871, −1.80636949032861795819643728766, −0.865440736491405725008687668011, −0.52071941934844237508838271620, 0.52071941934844237508838271620, 0.865440736491405725008687668011, 1.80636949032861795819643728766, 1.81038388398900894849820668871, 2.88897734265127925217414309474, 2.96148040917870694438810242911, 3.75246637682478132245336427694, 4.28148016975867483145426968632, 4.59647152603746605198860374674, 5.50476369418244116681712293780, 6.18841066958626046258365738347, 6.70625691975758057482531836782, 7.30796439538500621827271816730, 7.48621736765211734997499098291, 8.495997507737936948496271897935, 8.865676492525466583574453318746, 9.330698331600928852650419157591, 9.690470117714450293190299982649, 10.16615095017602300676912623779, 10.94436693500532443005117157072

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