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Properties

Label 4-48e2-1.1-c23e2-0-2
Degree $4$
Conductor $2304$
Sign $1$
Analytic cond. $25888.1$
Root an. cond. $12.6845$
Motivic weight $23$
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  + 3.54e5·3-s − 4.68e7·5-s + 2.11e8·7-s + 9.41e10·9-s − 1.46e12·11-s + 1.04e13·13-s − 1.65e13·15-s − 2.10e14·17-s + 9.07e14·19-s + 7.50e13·21-s − 1.01e16·23-s − 3.42e15·25-s + 2.22e16·27-s + 1.84e16·29-s + 2.72e17·31-s − 5.20e17·33-s − 9.92e15·35-s − 4.78e14·37-s + 3.71e18·39-s + 5.55e18·41-s + 1.19e18·43-s − 4.40e18·45-s − 2.63e19·47-s + 8.63e18·49-s − 7.47e19·51-s − 4.11e19·53-s + 6.87e19·55-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.428·5-s + 0.0405·7-s + 9-s − 1.55·11-s + 1.62·13-s − 0.495·15-s − 1.49·17-s + 1.78·19-s + 0.0467·21-s − 2.21·23-s − 0.286·25-s + 0.769·27-s + 0.281·29-s + 1.92·31-s − 1.79·33-s − 0.0173·35-s − 0.000442·37-s + 1.87·39-s + 1.57·41-s + 0.196·43-s − 0.428·45-s − 1.55·47-s + 0.315·49-s − 1.72·51-s − 0.609·53-s + 0.665·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $1$
Analytic conductor: \(25888.1\)
Root analytic conductor: \(12.6845\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2304,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(6.382851091\)
\(L(\frac12)\) \(\approx\) \(6.382851091\)
\(L(\frac{25}{2})\) 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$C_1$ \( ( 1 - p^{11} T )^{2} \)
good5$D_{4}$ \( 1 + 9361764 p T + 8977934193886 p^{4} T^{2} + 9361764 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 211963904 T - 175324586455056306 p^{2} T^{2} - 211963904 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 133542942408 p T + \)\(18\!\cdots\!54\)\( p^{2} T^{2} + 133542942408 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 10491654264748 T + \)\(79\!\cdots\!82\)\( p T^{2} - 10491654264748 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 12405177148284 p T + \)\(17\!\cdots\!98\)\( p^{2} T^{2} + 12405177148284 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 - 907382448537944 T + \)\(32\!\cdots\!42\)\( p T^{2} - 907382448537944 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 10116923323892112 T + \)\(67\!\cdots\!94\)\( T^{2} + 10116923323892112 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 637198398109548 p T + \)\(81\!\cdots\!38\)\( T^{2} - 637198398109548 p^{24} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 - 272793622592745488 T + \)\(55\!\cdots\!82\)\( T^{2} - 272793622592745488 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 478995036787364 T + \)\(91\!\cdots\!26\)\( T^{2} + 478995036787364 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 5555714961308771412 T + \)\(17\!\cdots\!22\)\( T^{2} - 5555714961308771412 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 - 1198322147609320040 T - \)\(25\!\cdots\!10\)\( T^{2} - 1198322147609320040 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 26308565672855777280 T + \)\(74\!\cdots\!90\)\( T^{2} + 26308565672855777280 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + 41127501899415224628 T + \)\(64\!\cdots\!74\)\( T^{2} + 41127501899415224628 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - \)\(30\!\cdots\!96\)\( T + \)\(11\!\cdots\!58\)\( T^{2} - \)\(30\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - \)\(59\!\cdots\!20\)\( T + \)\(26\!\cdots\!38\)\( T^{2} - \)\(59\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(16\!\cdots\!88\)\( T + \)\(24\!\cdots\!62\)\( T^{2} - \)\(16\!\cdots\!88\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(38\!\cdots\!76\)\( T + \)\(28\!\cdots\!66\)\( T^{2} - \)\(38\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(28\!\cdots\!28\)\( T + \)\(13\!\cdots\!54\)\( T^{2} + \)\(28\!\cdots\!28\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + 21069388575313284880 T + \)\(88\!\cdots\!78\)\( T^{2} + 21069388575313284880 p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(32\!\cdots\!42\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 + \)\(43\!\cdots\!64\)\( T + \)\(15\!\cdots\!18\)\( T^{2} + \)\(43\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(10\!\cdots\!92\)\( T + \)\(10\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!92\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−11.48073418441812039591988180687, −10.92160850552694720830548800209, −10.16979165056901625815070170480, −9.857591934590609998847470903188, −9.200418774869806978166330099075, −8.394265135060835281423887787339, −8.117977495772914286132385008763, −7.907374377206360660608684178491, −7.06905314055920625282912453763, −6.43862408548081759835405084129, −5.79677441338877949979480284066, −5.14337285484909851218050687429, −4.24266139368601656631184266605, −4.09329801952537671502642941657, −3.21574382092836948691378323913, −2.90372391454675358721395006982, −2.12201807870852333761733988358, −1.84092765587996876670884301618, −0.68510907766286219265041783219, −0.65931674201393174417122892785, 0.65931674201393174417122892785, 0.68510907766286219265041783219, 1.84092765587996876670884301618, 2.12201807870852333761733988358, 2.90372391454675358721395006982, 3.21574382092836948691378323913, 4.09329801952537671502642941657, 4.24266139368601656631184266605, 5.14337285484909851218050687429, 5.79677441338877949979480284066, 6.43862408548081759835405084129, 7.06905314055920625282912453763, 7.907374377206360660608684178491, 8.117977495772914286132385008763, 8.394265135060835281423887787339, 9.200418774869806978166330099075, 9.857591934590609998847470903188, 10.16979165056901625815070170480, 10.92160850552694720830548800209, 11.48073418441812039591988180687

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