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Properties

Label 4-10e2-1.1-c31e2-0-2
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $3706.02$
Root an. cond. $7.80237$
Motivic weight $31$
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  + 6.55e4·2-s + 3.89e7·3-s + 3.22e9·4-s − 6.10e10·5-s + 2.55e12·6-s − 1.20e13·7-s + 1.40e14·8-s + 1.49e14·9-s − 4.00e15·10-s − 3.98e15·11-s + 1.25e17·12-s + 5.81e15·13-s − 7.88e17·14-s − 2.37e18·15-s + 5.76e18·16-s − 9.08e16·17-s + 9.82e18·18-s − 9.10e19·19-s − 1.96e20·20-s − 4.68e20·21-s − 2.61e20·22-s − 1.66e21·23-s + 5.48e21·24-s + 2.79e21·25-s + 3.80e20·26-s − 2.33e22·27-s − 3.87e22·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.56·3-s + 3/2·4-s − 0.894·5-s + 2.21·6-s − 0.958·7-s + 1.41·8-s + 0.242·9-s − 1.26·10-s − 0.287·11-s + 2.35·12-s + 0.0314·13-s − 1.35·14-s − 1.40·15-s + 5/4·16-s − 0.00769·17-s + 0.343·18-s − 1.37·19-s − 1.34·20-s − 1.50·21-s − 0.407·22-s − 1.30·23-s + 2.21·24-s + 3/5·25-s + 0.0445·26-s − 1.51·27-s − 1.43·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(32-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+31/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(3706.02\)
Root analytic conductor: \(7.80237\)
Motivic weight: \(31\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{33}{2})\) 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$C_1$ \( ( 1 - p^{15} T )^{2} \)
5$C_1$ \( ( 1 + p^{15} T )^{2} \)
good3$D_{4}$ \( 1 - 4326484 p^{2} T + 624714725294 p^{7} T^{2} - 4326484 p^{33} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 + 245636353988 p^{2} T + \)\(22\!\cdots\!46\)\( p^{5} T^{2} + 245636353988 p^{33} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 3988310318131776 T + \)\(30\!\cdots\!06\)\( p T^{2} + 3988310318131776 p^{31} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 - 447180343089892 p T + \)\(29\!\cdots\!74\)\( p^{3} T^{2} - 447180343089892 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 + 90808009705448652 T + \)\(20\!\cdots\!42\)\( T^{2} + 90808009705448652 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 + 91080908739917057240 T + \)\(46\!\cdots\!02\)\( p T^{2} + 91080908739917057240 p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 + 72487748903732510868 p T + \)\(61\!\cdots\!82\)\( p^{2} T^{2} + 72487748903732510868 p^{32} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 + \)\(32\!\cdots\!40\)\( p T + \)\(77\!\cdots\!38\)\( p^{2} T^{2} + \)\(32\!\cdots\!40\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 + \)\(41\!\cdots\!56\)\( p T + \)\(36\!\cdots\!86\)\( T^{2} + \)\(41\!\cdots\!56\)\( p^{32} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(21\!\cdots\!32\)\( T + \)\(48\!\cdots\!82\)\( T^{2} + \)\(21\!\cdots\!32\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 + \)\(24\!\cdots\!76\)\( p T + \)\(13\!\cdots\!46\)\( T^{2} + \)\(24\!\cdots\!76\)\( p^{32} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(10\!\cdots\!16\)\( T + \)\(76\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!16\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 - \)\(10\!\cdots\!28\)\( T + \)\(85\!\cdots\!02\)\( T^{2} - \)\(10\!\cdots\!28\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(15\!\cdots\!56\)\( T + \)\(56\!\cdots\!78\)\( T^{2} - \)\(15\!\cdots\!56\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 + \)\(41\!\cdots\!20\)\( T - \)\(36\!\cdots\!82\)\( T^{2} + \)\(41\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 + \)\(10\!\cdots\!76\)\( T + \)\(59\!\cdots\!66\)\( T^{2} + \)\(10\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 + \)\(28\!\cdots\!52\)\( T + \)\(96\!\cdots\!42\)\( T^{2} + \)\(28\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 + \)\(50\!\cdots\!56\)\( T + \)\(32\!\cdots\!26\)\( T^{2} + \)\(50\!\cdots\!56\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 + \)\(14\!\cdots\!64\)\( T + \)\(13\!\cdots\!78\)\( T^{2} + \)\(14\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 + \)\(41\!\cdots\!60\)\( T + \)\(15\!\cdots\!58\)\( T^{2} + \)\(41\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 - \)\(81\!\cdots\!76\)\( T + \)\(48\!\cdots\!78\)\( T^{2} - \)\(81\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 - \)\(43\!\cdots\!20\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(43\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 - \)\(35\!\cdots\!28\)\( T + \)\(33\!\cdots\!02\)\( T^{2} - \)\(35\!\cdots\!28\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−13.25129642095814465644907896598, −13.22675274283704039825883465505, −12.29220750009867424823167517172, −11.70759852142898920607028920529, −10.83952189156692821548815250398, −10.13918560999091811341745471003, −8.938268789095556466354512330021, −8.680083224204161246746740425056, −7.50740521851980776096344695676, −7.44971581113566098046979031324, −6.15536446147803791253968916877, −5.77417643163608552608697863789, −4.58289123684851656691846756273, −3.93964003609424359527125062775, −3.29483301015647943691546324560, −3.16517723602498177324474627572, −2.14097577537524953912673940571, −1.83435827452413777695292052411, 0, 0, 1.83435827452413777695292052411, 2.14097577537524953912673940571, 3.16517723602498177324474627572, 3.29483301015647943691546324560, 3.93964003609424359527125062775, 4.58289123684851656691846756273, 5.77417643163608552608697863789, 6.15536446147803791253968916877, 7.44971581113566098046979031324, 7.50740521851980776096344695676, 8.680083224204161246746740425056, 8.938268789095556466354512330021, 10.13918560999091811341745471003, 10.83952189156692821548815250398, 11.70759852142898920607028920529, 12.29220750009867424823167517172, 13.22675274283704039825883465505, 13.25129642095814465644907896598

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