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Properties

Label 4-10e2-1.1-c23e2-0-2
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $1123.61$
Root an. cond. $5.78968$
Motivic weight $23$
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  + 4.09e3·2-s − 9.18e4·3-s + 1.25e7·4-s − 9.76e7·5-s − 3.76e8·6-s + 2.14e9·7-s + 3.43e10·8-s − 9.60e10·9-s − 4.00e11·10-s − 6.92e11·11-s − 1.15e12·12-s − 1.70e12·13-s + 8.79e12·14-s + 8.97e12·15-s + 8.79e13·16-s − 9.31e13·17-s − 3.93e14·18-s − 9.70e14·19-s − 1.22e15·20-s − 1.97e14·21-s − 2.83e15·22-s − 9.62e14·23-s − 3.15e15·24-s + 7.15e15·25-s − 6.99e15·26-s + 9.76e15·27-s + 2.70e16·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.299·3-s + 3/2·4-s − 0.894·5-s − 0.423·6-s + 0.410·7-s + 1.41·8-s − 1.01·9-s − 1.26·10-s − 0.731·11-s − 0.449·12-s − 0.264·13-s + 0.580·14-s + 0.267·15-s + 5/4·16-s − 0.659·17-s − 1.44·18-s − 1.91·19-s − 1.34·20-s − 0.122·21-s − 1.03·22-s − 0.210·23-s − 0.423·24-s + 3/5·25-s − 0.373·26-s + 0.338·27-s + 0.615·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+23/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1123.61\)
Root analytic conductor: \(5.78968\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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)$
bad2$C_1$ \( ( 1 - p^{11} T )^{2} \)
5$C_1$ \( ( 1 + p^{11} T )^{2} \)
good3$D_{4}$ \( 1 + 30628 p T + 429805726 p^{5} T^{2} + 30628 p^{24} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 - 306579844 p T + 101274343560173214 p^{3} T^{2} - 306579844 p^{24} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 + 62935415616 p T + \)\(15\!\cdots\!86\)\( p^{2} T^{2} + 62935415616 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 + 1708452301484 T + \)\(10\!\cdots\!66\)\( p T^{2} + 1708452301484 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 93149332240332 T + \)\(21\!\cdots\!46\)\( p T^{2} + 93149332240332 p^{23} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 970438534616600 T + \)\(38\!\cdots\!22\)\( p T^{2} + 970438534616600 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 + 962002498409964 T + \)\(31\!\cdots\!58\)\( T^{2} + 962002498409964 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 + 40394756391690180 T + \)\(26\!\cdots\!78\)\( T^{2} + 40394756391690180 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 + 54819002015519096 T + \)\(33\!\cdots\!86\)\( T^{2} + 54819002015519096 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 + 2098544042837784332 T + \)\(31\!\cdots\!62\)\( T^{2} + 2098544042837784332 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 + 4273739814696341076 T + \)\(22\!\cdots\!86\)\( T^{2} + 4273739814696341076 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 + 10838321168303261564 T + \)\(99\!\cdots\!38\)\( T^{2} + 10838321168303261564 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 + 26680537462560731892 T + \)\(52\!\cdots\!62\)\( T^{2} + 26680537462560731892 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - 33753250226251705956 T + \)\(57\!\cdots\!38\)\( T^{2} - 33753250226251705956 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 - 40199521927491576840 T + \)\(66\!\cdots\!58\)\( T^{2} - 40199521927491576840 p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 - \)\(80\!\cdots\!04\)\( T + \)\(37\!\cdots\!66\)\( T^{2} - \)\(80\!\cdots\!04\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(21\!\cdots\!48\)\( T + \)\(29\!\cdots\!02\)\( T^{2} - \)\(21\!\cdots\!48\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(71\!\cdots\!64\)\( T - \)\(15\!\cdots\!54\)\( T^{2} - \)\(71\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 - \)\(31\!\cdots\!76\)\( T + \)\(90\!\cdots\!78\)\( T^{2} - \)\(31\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + \)\(70\!\cdots\!20\)\( T + \)\(10\!\cdots\!78\)\( T^{2} + \)\(70\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 + \)\(21\!\cdots\!44\)\( T + \)\(32\!\cdots\!58\)\( T^{2} + \)\(21\!\cdots\!44\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 + \)\(19\!\cdots\!80\)\( T + \)\(11\!\cdots\!38\)\( T^{2} + \)\(19\!\cdots\!80\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(11\!\cdots\!88\)\( T + \)\(10\!\cdots\!82\)\( T^{2} - \)\(11\!\cdots\!88\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−14.55430448726141694530821031557, −14.52714145053782251705922784833, −13.20883757430877632738312681175, −12.92354923669322687883018516492, −11.95917950019260127343796694263, −11.43494994566668803452833107465, −10.94698648401794219119414465327, −10.13369762318517327652199063516, −8.392302418733245324373918091481, −8.288694230353171644476465211036, −6.94103249727586938170408291690, −6.49645296786283932791073601534, −5.23431630705502910321398833247, −5.06211035121655743552037750147, −3.98167045682760011718907147567, −3.39131263719406303899277868045, −2.39426470800482057139001456287, −1.75844881679893128551930352218, 0, 0, 1.75844881679893128551930352218, 2.39426470800482057139001456287, 3.39131263719406303899277868045, 3.98167045682760011718907147567, 5.06211035121655743552037750147, 5.23431630705502910321398833247, 6.49645296786283932791073601534, 6.94103249727586938170408291690, 8.288694230353171644476465211036, 8.392302418733245324373918091481, 10.13369762318517327652199063516, 10.94698648401794219119414465327, 11.43494994566668803452833107465, 11.95917950019260127343796694263, 12.92354923669322687883018516492, 13.20883757430877632738312681175, 14.52714145053782251705922784833, 14.55430448726141694530821031557

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