↑

Properties

Label 4-2e2-1.1-c45e2-0-0
Degree $4$
Conductor $4$
Sign $1$
Analytic cond. $657.981$
Root an. cond. $5.06469$
Motivic weight $45$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8.38e6·2-s + 5.98e10·3-s + 5.27e13·4-s + 4.39e15·5-s + 5.02e17·6-s + 7.90e18·7-s + 2.95e20·8-s − 2.69e20·9-s + 3.68e22·10-s + 1.93e23·11-s + 3.15e24·12-s − 2.31e24·13-s + 6.62e25·14-s + 2.63e26·15-s + 1.54e27·16-s − 6.57e27·17-s − 2.26e27·18-s − 9.00e28·19-s + 2.32e29·20-s + 4.73e29·21-s + 1.62e30·22-s + 6.99e30·23-s + 1.76e31·24-s + 6.00e30·25-s − 1.93e31·26-s − 6.99e31·27-s + 4.17e32·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.10·3-s + 3/2·4-s + 0.824·5-s + 1.55·6-s + 0.763·7-s + 1.41·8-s − 0.0912·9-s + 1.16·10-s + 0.716·11-s + 1.65·12-s − 0.199·13-s + 1.08·14-s + 0.908·15-s + 5/4·16-s − 1.35·17-s − 0.128·18-s − 1.52·19-s + 1.23·20-s + 0.841·21-s + 1.01·22-s + 1.60·23-s + 1.55·24-s + 0.211·25-s − 0.282·26-s − 0.435·27-s + 1.14·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(46-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+45/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(657.981\)
Root analytic conductor: \(5.06469\)
Motivic weight: \(45\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4,\ (\ :45/2, 45/2),\ 1)\)

Particular Values

\(L(23)\) \(\approx\) \(15.34386415\)
\(L(\frac12)\) \(\approx\) \(15.34386415\)
\(L(\frac{47}{2})\) 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$C_1$ \( ( 1 - p^{22} T )^{2} \)
good3$D_{4}$ \( 1 - 19953739064 p T + 195744721333750994 p^{9} T^{2} - 19953739064 p^{46} T^{3} + p^{90} T^{4} \)
5$D_{4}$ \( 1 - 175884559656204 p^{2} T + \)\(34\!\cdots\!22\)\( p^{8} T^{2} - 175884559656204 p^{47} T^{3} + p^{90} T^{4} \)
7$D_{4}$ \( 1 - 1128927599067328432 p T + \)\(13\!\cdots\!94\)\( p^{5} T^{2} - 1128927599067328432 p^{46} T^{3} + p^{90} T^{4} \)
11$D_{4}$ \( 1 - \)\(17\!\cdots\!04\)\( p T + \)\(10\!\cdots\!06\)\( p^{3} T^{2} - \)\(17\!\cdots\!04\)\( p^{46} T^{3} + p^{90} T^{4} \)
13$D_{4}$ \( 1 + \)\(17\!\cdots\!56\)\( p T + \)\(35\!\cdots\!62\)\( p^{4} T^{2} + \)\(17\!\cdots\!56\)\( p^{46} T^{3} + p^{90} T^{4} \)
17$D_{4}$ \( 1 + \)\(65\!\cdots\!36\)\( T + \)\(11\!\cdots\!42\)\( p^{2} T^{2} + \)\(65\!\cdots\!36\)\( p^{45} T^{3} + p^{90} T^{4} \)
19$D_{4}$ \( 1 + \)\(90\!\cdots\!20\)\( T + \)\(19\!\cdots\!18\)\( p^{2} T^{2} + \)\(90\!\cdots\!20\)\( p^{45} T^{3} + p^{90} T^{4} \)
23$D_{4}$ \( 1 - \)\(30\!\cdots\!44\)\( p T + \)\(72\!\cdots\!18\)\( p^{2} T^{2} - \)\(30\!\cdots\!44\)\( p^{46} T^{3} + p^{90} T^{4} \)
29$D_{4}$ \( 1 - \)\(76\!\cdots\!60\)\( p T + \)\(29\!\cdots\!78\)\( p^{2} T^{2} - \)\(76\!\cdots\!60\)\( p^{46} T^{3} + p^{90} T^{4} \)
31$D_{4}$ \( 1 - \)\(31\!\cdots\!64\)\( p T + \)\(48\!\cdots\!06\)\( p^{2} T^{2} - \)\(31\!\cdots\!64\)\( p^{46} T^{3} + p^{90} T^{4} \)
37$D_{4}$ \( 1 + \)\(51\!\cdots\!08\)\( p T + \)\(60\!\cdots\!22\)\( p^{2} T^{2} + \)\(51\!\cdots\!08\)\( p^{46} T^{3} + p^{90} T^{4} \)
41$D_{4}$ \( 1 + \)\(45\!\cdots\!36\)\( p T + \)\(37\!\cdots\!66\)\( p^{2} T^{2} + \)\(45\!\cdots\!36\)\( p^{46} T^{3} + p^{90} T^{4} \)
43$D_{4}$ \( 1 + \)\(18\!\cdots\!96\)\( p T + \)\(19\!\cdots\!18\)\( p^{2} T^{2} + \)\(18\!\cdots\!96\)\( p^{46} T^{3} + p^{90} T^{4} \)
47$D_{4}$ \( 1 - \)\(36\!\cdots\!04\)\( T + \)\(38\!\cdots\!18\)\( T^{2} - \)\(36\!\cdots\!04\)\( p^{45} T^{3} + p^{90} T^{4} \)
53$D_{4}$ \( 1 + \)\(66\!\cdots\!88\)\( T + \)\(20\!\cdots\!22\)\( T^{2} + \)\(66\!\cdots\!88\)\( p^{45} T^{3} + p^{90} T^{4} \)
59$D_{4}$ \( 1 + \)\(14\!\cdots\!20\)\( T + \)\(53\!\cdots\!98\)\( T^{2} + \)\(14\!\cdots\!20\)\( p^{45} T^{3} + p^{90} T^{4} \)
61$D_{4}$ \( 1 - \)\(16\!\cdots\!64\)\( T + \)\(42\!\cdots\!26\)\( T^{2} - \)\(16\!\cdots\!64\)\( p^{45} T^{3} + p^{90} T^{4} \)
67$D_{4}$ \( 1 + \)\(82\!\cdots\!76\)\( T + \)\(22\!\cdots\!58\)\( T^{2} + \)\(82\!\cdots\!76\)\( p^{45} T^{3} + p^{90} T^{4} \)
71$D_{4}$ \( 1 - \)\(11\!\cdots\!64\)\( T + \)\(23\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!64\)\( p^{45} T^{3} + p^{90} T^{4} \)
73$D_{4}$ \( 1 - \)\(13\!\cdots\!32\)\( T + \)\(15\!\cdots\!42\)\( T^{2} - \)\(13\!\cdots\!32\)\( p^{45} T^{3} + p^{90} T^{4} \)
79$D_{4}$ \( 1 - \)\(33\!\cdots\!00\)\( T + \)\(51\!\cdots\!98\)\( T^{2} - \)\(33\!\cdots\!00\)\( p^{45} T^{3} + p^{90} T^{4} \)
83$D_{4}$ \( 1 + \)\(95\!\cdots\!48\)\( T + \)\(32\!\cdots\!62\)\( T^{2} + \)\(95\!\cdots\!48\)\( p^{45} T^{3} + p^{90} T^{4} \)
89$D_{4}$ \( 1 - \)\(67\!\cdots\!80\)\( T + \)\(11\!\cdots\!98\)\( T^{2} - \)\(67\!\cdots\!80\)\( p^{45} T^{3} + p^{90} T^{4} \)
97$D_{4}$ \( 1 + \)\(82\!\cdots\!36\)\( T + \)\(67\!\cdots\!38\)\( T^{2} + \)\(82\!\cdots\!36\)\( p^{45} T^{3} + p^{90} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.47181041753520800793968944156, −17.22155488884793515637000171613, −15.59728930414843338272652625574, −15.02998543214135205344843893413, −14.07006072227554698187630433643, −13.96916314935017875788821637951, −13.02465815304419359437668139808, −12.02335491260035703668570003039, −11.08295708198227584361145440251, −10.13036915167214385632441297663, −8.699816738451365361174609907510, −8.339124232919859230619858525948, −6.61719473087070625099122866796, −6.43773033268690564252243006983, −4.76977040760399375733466440754, −4.64959870563031222004499275381, −3.25545818777260741480786656395, −2.56767703649441236695384257121, −2.01162741302255434184756689084, −1.00239020039170817281045207556, 1.00239020039170817281045207556, 2.01162741302255434184756689084, 2.56767703649441236695384257121, 3.25545818777260741480786656395, 4.64959870563031222004499275381, 4.76977040760399375733466440754, 6.43773033268690564252243006983, 6.61719473087070625099122866796, 8.339124232919859230619858525948, 8.699816738451365361174609907510, 10.13036915167214385632441297663, 11.08295708198227584361145440251, 12.02335491260035703668570003039, 13.02465815304419359437668139808, 13.96916314935017875788821637951, 14.07006072227554698187630433643, 15.02998543214135205344843893413, 15.59728930414843338272652625574, 17.22155488884793515637000171613, 17.47181041753520800793968944156

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