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Properties

Label 4-10e2-1.1-c31e2-0-0
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.55e4·2-s + 2.90e7·3-s + 3.22e9·4-s − 6.10e10·5-s − 1.90e12·6-s + 1.27e13·7-s − 1.40e14·8-s − 5.89e14·9-s + 4.00e15·10-s − 5.00e14·11-s + 9.34e16·12-s − 1.99e17·13-s − 8.38e17·14-s − 1.77e18·15-s + 5.76e18·16-s + 1.21e19·17-s + 3.86e19·18-s + 1.27e20·19-s − 1.96e20·20-s + 3.71e20·21-s + 3.28e19·22-s + 1.26e21·23-s − 4.08e21·24-s + 2.79e21·25-s + 1.30e22·26-s − 4.07e22·27-s + 4.12e22·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.16·3-s + 3/2·4-s − 0.894·5-s − 1.65·6-s + 1.01·7-s − 1.41·8-s − 0.954·9-s + 1.26·10-s − 0.0361·11-s + 1.75·12-s − 1.08·13-s − 1.44·14-s − 1.04·15-s + 5/4·16-s + 1.03·17-s + 1.34·18-s + 1.92·19-s − 1.34·20-s + 1.18·21-s + 0.0511·22-s + 0.990·23-s − 1.65·24-s + 3/5·25-s + 1.52·26-s − 2.65·27-s + 1.52·28-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: \(0\)
Selberg data: \((4,\ 100,\ (\ :31/2, 31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(2.035694691\)
\(L(\frac12)\) \(\approx\) \(2.035694691\)
\(L(\frac{33}{2})\) 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$D_{4}$ \( 1 - 358324 p^{4} T + 24247506922 p^{10} T^{2} - 358324 p^{35} T^{3} + p^{62} T^{4} \)
7$D_{4}$ \( 1 - 1827879729316 p T + \)\(39\!\cdots\!22\)\( p^{4} T^{2} - 1827879729316 p^{32} T^{3} + p^{62} T^{4} \)
11$D_{4}$ \( 1 + 500906012122176 T - \)\(20\!\cdots\!94\)\( p T^{2} + 500906012122176 p^{31} T^{3} + p^{62} T^{4} \)
13$D_{4}$ \( 1 + 15341543253854492 p T + \)\(44\!\cdots\!62\)\( p^{2} T^{2} + 15341543253854492 p^{32} T^{3} + p^{62} T^{4} \)
17$D_{4}$ \( 1 - 12159060327494700852 T + \)\(16\!\cdots\!26\)\( p T^{2} - 12159060327494700852 p^{31} T^{3} + p^{62} T^{4} \)
19$D_{4}$ \( 1 - \)\(12\!\cdots\!60\)\( T + \)\(63\!\cdots\!02\)\( p T^{2} - \)\(12\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \)
23$D_{4}$ \( 1 - \)\(12\!\cdots\!64\)\( T + \)\(15\!\cdots\!86\)\( p T^{2} - \)\(12\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
29$D_{4}$ \( 1 + \)\(31\!\cdots\!40\)\( p T - \)\(26\!\cdots\!62\)\( p^{2} T^{2} + \)\(31\!\cdots\!40\)\( p^{32} T^{3} + p^{62} T^{4} \)
31$D_{4}$ \( 1 + \)\(14\!\cdots\!36\)\( T + \)\(30\!\cdots\!86\)\( T^{2} + \)\(14\!\cdots\!36\)\( p^{31} T^{3} + p^{62} T^{4} \)
37$D_{4}$ \( 1 + \)\(47\!\cdots\!68\)\( T + \)\(13\!\cdots\!82\)\( T^{2} + \)\(47\!\cdots\!68\)\( p^{31} T^{3} + p^{62} T^{4} \)
41$D_{4}$ \( 1 - \)\(14\!\cdots\!84\)\( T + \)\(24\!\cdots\!46\)\( T^{2} - \)\(14\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
43$D_{4}$ \( 1 - \)\(28\!\cdots\!84\)\( T + \)\(10\!\cdots\!78\)\( T^{2} - \)\(28\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \)
47$D_{4}$ \( 1 + \)\(51\!\cdots\!28\)\( T + \)\(88\!\cdots\!02\)\( T^{2} + \)\(51\!\cdots\!28\)\( p^{31} T^{3} + p^{62} T^{4} \)
53$D_{4}$ \( 1 - \)\(61\!\cdots\!44\)\( T + \)\(61\!\cdots\!78\)\( T^{2} - \)\(61\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
59$D_{4}$ \( 1 - \)\(54\!\cdots\!80\)\( T + \)\(22\!\cdots\!18\)\( T^{2} - \)\(54\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
61$D_{4}$ \( 1 - \)\(28\!\cdots\!24\)\( T + \)\(19\!\cdots\!66\)\( T^{2} - \)\(28\!\cdots\!24\)\( p^{31} T^{3} + p^{62} T^{4} \)
67$D_{4}$ \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(57\!\cdots\!42\)\( T^{2} - \)\(10\!\cdots\!52\)\( p^{31} T^{3} + p^{62} T^{4} \)
71$D_{4}$ \( 1 - \)\(58\!\cdots\!44\)\( T + \)\(57\!\cdots\!26\)\( T^{2} - \)\(58\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \)
73$D_{4}$ \( 1 - \)\(17\!\cdots\!64\)\( T + \)\(15\!\cdots\!78\)\( T^{2} - \)\(17\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \)
79$D_{4}$ \( 1 - \)\(18\!\cdots\!40\)\( T + \)\(33\!\cdots\!58\)\( T^{2} - \)\(18\!\cdots\!40\)\( p^{31} T^{3} + p^{62} T^{4} \)
83$D_{4}$ \( 1 - \)\(53\!\cdots\!24\)\( T + \)\(41\!\cdots\!78\)\( T^{2} - \)\(53\!\cdots\!24\)\( p^{31} T^{3} + p^{62} T^{4} \)
89$D_{4}$ \( 1 + \)\(75\!\cdots\!80\)\( T + \)\(31\!\cdots\!78\)\( T^{2} + \)\(75\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \)
97$D_{4}$ \( 1 + \)\(42\!\cdots\!28\)\( T + \)\(81\!\cdots\!02\)\( T^{2} + \)\(42\!\cdots\!28\)\( p^{31} T^{3} + p^{62} T^{4} \)
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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

−14.43001746334773705877343598807, −13.84203614138844073083266180466, −12.41105371858153765749923078374, −11.87716856148701995697642440913, −11.24952737001225955935701874864, −10.77638553179234885648872178839, −9.508695960247308912957377546135, −9.344886200642484944198761888823, −8.285495702833957877442910728068, −8.249202534134590961034378240617, −7.27982344695646936241333726631, −7.26941571540351477127699767725, −5.41981212374271320712452928847, −5.30216774150479921347713117739, −3.65887322428048848758089559006, −3.27712225625199937327068013358, −2.48150467550552921887619836060, −1.98951857919564664797491242082, −0.912107971976522102988485187780, −0.52924740275238647642561331536, 0.52924740275238647642561331536, 0.912107971976522102988485187780, 1.98951857919564664797491242082, 2.48150467550552921887619836060, 3.27712225625199937327068013358, 3.65887322428048848758089559006, 5.30216774150479921347713117739, 5.41981212374271320712452928847, 7.26941571540351477127699767725, 7.27982344695646936241333726631, 8.249202534134590961034378240617, 8.285495702833957877442910728068, 9.344886200642484944198761888823, 9.508695960247308912957377546135, 10.77638553179234885648872178839, 11.24952737001225955935701874864, 11.87716856148701995697642440913, 12.41105371858153765749923078374, 13.84203614138844073083266180466, 14.43001746334773705877343598807

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