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Properties

Label 4-5e4-1.1-c23e2-0-0
Degree $4$
Conductor $625$
Sign $1$
Analytic cond. $7022.60$
Root an. cond. $9.15428$
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  − 1.08e3·2-s − 3.39e5·3-s + 4.85e6·4-s + 3.66e8·6-s + 1.35e9·7-s − 1.82e10·8-s − 5.40e10·9-s + 8.56e11·11-s − 1.64e12·12-s − 4.37e12·13-s − 1.46e12·14-s − 2.28e13·16-s − 2.54e14·17-s + 5.83e13·18-s + 4.26e12·19-s − 4.61e14·21-s − 9.25e14·22-s + 8.14e15·23-s + 6.21e15·24-s + 4.72e15·26-s + 4.38e16·27-s + 6.60e15·28-s + 2.08e16·29-s + 1.37e17·31-s − 1.08e16·32-s − 2.90e17·33-s + 2.74e17·34-s + ⋯
L(s)  = 1  − 0.372·2-s − 1.10·3-s + 0.579·4-s + 0.412·6-s + 0.259·7-s − 0.752·8-s − 0.573·9-s + 0.905·11-s − 0.640·12-s − 0.677·13-s − 0.0968·14-s − 0.325·16-s − 1.79·17-s + 0.213·18-s + 0.00839·19-s − 0.287·21-s − 0.337·22-s + 1.78·23-s + 0.833·24-s + 0.252·26-s + 1.51·27-s + 0.150·28-s + 0.316·29-s + 0.973·31-s − 0.0530·32-s − 1.00·33-s + 0.670·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(625\)    =    \(5^{4}\)
Sign: $1$
Analytic conductor: \(7022.60\)
Root analytic conductor: \(9.15428\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 625,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(1.565456336\)
\(L(\frac12)\) \(\approx\) \(1.565456336\)
\(L(\frac{25}{2})\) 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)$
bad5 \( 1 \)
good2$D_{4}$ \( 1 + 135 p^{3} T - 3605 p^{10} T^{2} + 135 p^{26} T^{3} + p^{46} T^{4} \)
3$D_{4}$ \( 1 + 37720 p^{2} T + 77396530 p^{7} T^{2} + 37720 p^{25} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 194169200 p T + 102109123349477250 p^{3} T^{2} - 194169200 p^{24} T^{3} + p^{46} T^{4} \)
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

−12.74582234142462004227932436701, −11.90193002020823932339913474313, −11.61165657927483258112748298807, −11.19915736248966531597929686851, −10.76210504882136473605865528192, −9.855702876871078000352675485762, −8.971196529638716008857195859792, −8.849073239713488833418643593286, −7.921785982507148123834097495642, −6.86484226012888738802830151210, −6.58602291235129800034266331427, −6.19248536121156329932400571951, −5.11990928209704657605271777585, −4.89612160098352117364581473701, −3.95997718386367537560736170234, −2.90583598968921865551321300499, −2.48577793966964915075129149878, −1.73365832574989764201565941153, −0.62933366023716082329223072198, −0.54623411542559162509864494023, 0.54623411542559162509864494023, 0.62933366023716082329223072198, 1.73365832574989764201565941153, 2.48577793966964915075129149878, 2.90583598968921865551321300499, 3.95997718386367537560736170234, 4.89612160098352117364581473701, 5.11990928209704657605271777585, 6.19248536121156329932400571951, 6.58602291235129800034266331427, 6.86484226012888738802830151210, 7.921785982507148123834097495642, 8.849073239713488833418643593286, 8.971196529638716008857195859792, 9.855702876871078000352675485762, 10.76210504882136473605865528192, 11.19915736248966531597929686851, 11.61165657927483258112748298807, 11.90193002020823932339913474313, 12.74582234142462004227932436701

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