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Properties

Label 4-7e4-1.1-c23e2-0-0
Degree $4$
Conductor $2401$
Sign $1$
Analytic cond. $26978.0$
Root an. cond. $12.8160$
Motivic weight $23$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.08e3·2-s − 3.39e5·3-s + 4.85e6·4-s − 7.30e7·5-s − 3.66e8·6-s + 1.82e10·8-s − 5.40e10·9-s − 7.89e10·10-s + 8.56e11·11-s − 1.64e12·12-s − 4.37e12·13-s + 2.48e13·15-s − 2.28e13·16-s − 2.54e14·17-s − 5.83e13·18-s − 4.26e12·19-s − 3.54e14·20-s + 9.25e14·22-s − 8.14e15·23-s − 6.21e15·24-s − 1.51e16·25-s − 4.72e15·26-s + 4.38e16·27-s + 2.08e16·29-s + 2.67e16·30-s − 1.37e17·31-s + 1.08e16·32-s + ⋯
L(s)  = 1  + 0.372·2-s − 1.10·3-s + 0.579·4-s − 0.669·5-s − 0.412·6-s + 0.752·8-s − 0.573·9-s − 0.249·10-s + 0.905·11-s − 0.640·12-s − 0.677·13-s + 0.740·15-s − 0.325·16-s − 1.79·17-s − 0.213·18-s − 0.00839·19-s − 0.387·20-s + 0.337·22-s − 1.78·23-s − 0.833·24-s − 1.27·25-s − 0.252·26-s + 1.51·27-s + 0.316·29-s + 0.276·30-s − 0.973·31-s + 0.0530·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $1$
Analytic conductor: \(26978.0\)
Root analytic conductor: \(12.8160\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2401,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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)$
bad7 \( 1 \)
good2$D_{4}$ \( 1 - 135 p^{3} T - 3605 p^{10} T^{2} - 135 p^{26} T^{3} + p^{46} T^{4} \)
3$D_{4}$ \( 1 + 37720 p^{2} T + 77396530 p^{7} T^{2} + 37720 p^{25} T^{3} + p^{46} T^{4} \)
5$D_{4}$ \( 1 + 14613804 p T + 32768971378174 p^{4} T^{2} + 14613804 p^{24} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 - 77891088024 p T + \)\(16\!\cdots\!66\)\( p^{2} T^{2} - 77891088024 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 + 4376109322060 T + \)\(42\!\cdots\!90\)\( p T^{2} + 4376109322060 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 14942832211620 p T + \)\(15\!\cdots\!10\)\( p^{2} T^{2} + 14942832211620 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 224242156840 p T + \)\(34\!\cdots\!38\)\( p^{2} T^{2} + 224242156840 p^{24} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 8144713079008560 T + \)\(58\!\cdots\!30\)\( T^{2} + 8144713079008560 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 20818433601623340 T + \)\(77\!\cdots\!78\)\( T^{2} - 20818433601623340 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 137714017177000384 T + \)\(40\!\cdots\!46\)\( T^{2} + 137714017177000384 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 897721264408967780 T + \)\(19\!\cdots\!70\)\( T^{2} + 897721264408967780 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 2294435477168314956 T + \)\(19\!\cdots\!26\)\( T^{2} - 2294435477168314956 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 1750760768619855800 T + \)\(73\!\cdots\!50\)\( T^{2} + 1750760768619855800 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 15759744217656780960 T + \)\(36\!\cdots\!10\)\( T^{2} + 15759744217656780960 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(13\!\cdots\!10\)\( T^{2} + \)\(14\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 + \)\(28\!\cdots\!80\)\( T + \)\(10\!\cdots\!58\)\( T^{2} + \)\(28\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - \)\(18\!\cdots\!36\)\( T + \)\(96\!\cdots\!86\)\( T^{2} - \)\(18\!\cdots\!36\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(17\!\cdots\!40\)\( T + \)\(24\!\cdots\!90\)\( T^{2} - \)\(17\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(30\!\cdots\!24\)\( T + \)\(93\!\cdots\!66\)\( T^{2} - \)\(30\!\cdots\!24\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 - \)\(80\!\cdots\!60\)\( T + \)\(30\!\cdots\!30\)\( T^{2} - \)\(80\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 - \)\(62\!\cdots\!40\)\( T + \)\(47\!\cdots\!78\)\( T^{2} - \)\(62\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 + \)\(68\!\cdots\!20\)\( T + \)\(16\!\cdots\!90\)\( T^{2} + \)\(68\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 + \)\(63\!\cdots\!20\)\( T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(63\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(31\!\cdots\!40\)\( T + \)\(53\!\cdots\!10\)\( T^{2} - \)\(31\!\cdots\!40\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−11.11015032111558489675308376640, −10.81591486587553546815721792259, −9.660153982755381791843747225969, −9.468198422406310202469890816024, −8.414063038711063383699037278229, −8.092913542311568983501425954333, −7.34432700308319776646433506223, −6.65346793454703319389046206466, −6.38792853132450593755718084761, −5.88234460918325334312416363161, −4.98420642922831133984871789837, −4.79462154288247396879038326801, −3.83749689703328734395550250741, −3.77607373681917342487380261656, −2.66336686533642895608001058774, −1.94185589552580867628957426830, −1.85629561139771802680929282611, −0.67599019628767350974839327353, 0, 0, 0.67599019628767350974839327353, 1.85629561139771802680929282611, 1.94185589552580867628957426830, 2.66336686533642895608001058774, 3.77607373681917342487380261656, 3.83749689703328734395550250741, 4.79462154288247396879038326801, 4.98420642922831133984871789837, 5.88234460918325334312416363161, 6.38792853132450593755718084761, 6.65346793454703319389046206466, 7.34432700308319776646433506223, 8.092913542311568983501425954333, 8.414063038711063383699037278229, 9.468198422406310202469890816024, 9.660153982755381791843747225969, 10.81591486587553546815721792259, 11.11015032111558489675308376640

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