↑

Properties

Label 4-10e2-1.1-c27e2-0-1
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $2133.10$
Root an. cond. $6.79599$
Motivic weight $27$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.63e4·2-s − 3.34e6·3-s + 2.01e8·4-s + 2.44e9·5-s + 5.47e10·6-s − 5.87e10·7-s − 2.19e12·8-s − 3.43e12·9-s − 4.00e13·10-s − 1.48e14·11-s − 6.72e14·12-s + 7.00e14·13-s + 9.61e14·14-s − 8.15e15·15-s + 2.25e16·16-s + 3.83e16·17-s + 5.62e16·18-s + 2.31e17·19-s + 4.91e17·20-s + 1.96e17·21-s + 2.43e18·22-s − 1.05e18·23-s + 7.34e18·24-s + 4.47e18·25-s − 1.14e19·26-s + 3.47e19·27-s − 1.18e19·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.20·3-s + 3/2·4-s + 0.894·5-s + 1.71·6-s − 0.229·7-s − 1.41·8-s − 0.450·9-s − 1.26·10-s − 1.29·11-s − 1.81·12-s + 0.641·13-s + 0.323·14-s − 1.08·15-s + 5/4·16-s + 0.939·17-s + 0.636·18-s + 1.26·19-s + 1.34·20-s + 0.277·21-s + 1.83·22-s − 0.435·23-s + 1.71·24-s + 3/5·25-s − 0.906·26-s + 1.64·27-s − 0.343·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2133.10\)
Root analytic conductor: \(6.79599\)
Motivic weight: \(27\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :27/2, 27/2),\ 1)\)

Particular Values

\(L(14)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{29}{2})\) 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$C_1$ \( ( 1 + p^{13} T )^{2} \)
5$C_1$ \( ( 1 - p^{13} T )^{2} \)
good3$D_{4}$ \( 1 + 1113548 p T + 60050290706 p^{5} T^{2} + 1113548 p^{28} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 8386691756 p T + \)\(20\!\cdots\!98\)\( p^{2} T^{2} + 8386691756 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 + 1227968656176 p^{2} T + \)\(11\!\cdots\!66\)\( p^{3} T^{2} + 1227968656176 p^{29} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 - 53853778386892 p T + \)\(45\!\cdots\!02\)\( p^{2} T^{2} - 53853778386892 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 - 38363966708465748 T + \)\(17\!\cdots\!66\)\( p T^{2} - 38363966708465748 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 - 12161973113018200 p T + \)\(14\!\cdots\!98\)\( p^{2} T^{2} - 12161973113018200 p^{28} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 1051524217202377644 T + \)\(14\!\cdots\!86\)\( p T^{2} + 1051524217202377644 p^{27} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 - 59213295656400885420 T + \)\(63\!\cdots\!18\)\( T^{2} - 59213295656400885420 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(27\!\cdots\!06\)\( T^{2} - \)\(10\!\cdots\!44\)\( p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 + 93534665260879182692 T - \)\(45\!\cdots\!18\)\( T^{2} + 93534665260879182692 p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 + \)\(10\!\cdots\!56\)\( T + \)\(95\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!56\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 + \)\(47\!\cdots\!64\)\( T + \)\(25\!\cdots\!38\)\( T^{2} + \)\(47\!\cdots\!64\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 + \)\(12\!\cdots\!72\)\( T + \)\(27\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!72\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 + \)\(10\!\cdots\!04\)\( T + \)\(45\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!04\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 + \)\(18\!\cdots\!60\)\( T + \)\(21\!\cdots\!38\)\( T^{2} + \)\(18\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 + \)\(37\!\cdots\!16\)\( T + \)\(66\!\cdots\!06\)\( T^{2} + \)\(37\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 - \)\(37\!\cdots\!28\)\( T + \)\(40\!\cdots\!42\)\( T^{2} - \)\(37\!\cdots\!28\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 + \)\(48\!\cdots\!76\)\( T + \)\(19\!\cdots\!26\)\( T^{2} + \)\(48\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 - \)\(94\!\cdots\!96\)\( T + \)\(32\!\cdots\!98\)\( T^{2} - \)\(94\!\cdots\!96\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 - \)\(91\!\cdots\!80\)\( T + \)\(42\!\cdots\!42\)\( p T^{2} - \)\(91\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(14\!\cdots\!16\)\( T + \)\(11\!\cdots\!18\)\( T^{2} - \)\(14\!\cdots\!16\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 + \)\(16\!\cdots\!80\)\( T + \)\(92\!\cdots\!58\)\( T^{2} + \)\(16\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 + \)\(88\!\cdots\!36\)\( p T + \)\(10\!\cdots\!42\)\( T^{2} + \)\(88\!\cdots\!36\)\( p^{28} T^{3} + p^{54} T^{4} \)
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

−13.97756821922551499166121253593, −13.49467399399764701293582925178, −12.08816433248692243662481854452, −12.07522271487377906900866341107, −10.82351813602922164737248597228, −10.72478428543871405819678904805, −9.840799220441137020884259230077, −9.349324796154985816624834683360, −8.190493865173437379062652528635, −7.926125608269574303601018424469, −6.61010164811198320189881389167, −6.23138509488249428180098377454, −5.44049308993059938142364929025, −5.03604025707529726661101869382, −3.05300331982851093961720436755, −2.88574255606246008863666505496, −1.56573529971417964589857727623, −1.13126610315316071062535865744, 0, 0, 1.13126610315316071062535865744, 1.56573529971417964589857727623, 2.88574255606246008863666505496, 3.05300331982851093961720436755, 5.03604025707529726661101869382, 5.44049308993059938142364929025, 6.23138509488249428180098377454, 6.61010164811198320189881389167, 7.926125608269574303601018424469, 8.190493865173437379062652528635, 9.349324796154985816624834683360, 9.840799220441137020884259230077, 10.72478428543871405819678904805, 10.82351813602922164737248597228, 12.07522271487377906900866341107, 12.08816433248692243662481854452, 13.49467399399764701293582925178, 13.97756821922551499166121253593

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