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Properties

Label 4-10e2-1.1-c21e2-0-2
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $781.075$
Root an. cond. $5.28656$
Motivic weight $21$
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  − 2.04e3·2-s − 1.26e5·3-s + 3.14e6·4-s − 1.95e7·5-s + 2.59e8·6-s − 2.92e8·7-s − 4.29e9·8-s + 3.49e9·9-s + 4.00e10·10-s − 4.13e10·11-s − 3.98e11·12-s + 1.88e12·13-s + 5.99e11·14-s + 2.47e12·15-s + 5.49e12·16-s − 3.10e12·17-s − 7.16e12·18-s + 7.17e12·19-s − 6.14e13·20-s + 3.70e13·21-s + 8.46e13·22-s − 3.57e14·23-s + 5.44e14·24-s + 2.86e14·25-s − 3.86e15·26-s − 1.78e14·27-s − 9.20e14·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.23·3-s + 3/2·4-s − 0.894·5-s + 1.75·6-s − 0.391·7-s − 1.41·8-s + 0.334·9-s + 1.26·10-s − 0.480·11-s − 1.85·12-s + 3.79·13-s + 0.553·14-s + 1.10·15-s + 5/4·16-s − 0.372·17-s − 0.472·18-s + 0.268·19-s − 1.34·20-s + 0.484·21-s + 0.679·22-s − 1.79·23-s + 1.75·24-s + 3/5·25-s − 5.36·26-s − 0.166·27-s − 0.587·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+21/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(781.075\)
Root analytic conductor: \(5.28656\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) 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$C_1$ \( ( 1 + p^{10} T )^{2} \)
5$C_1$ \( ( 1 + p^{10} T )^{2} \)
good3$D_{4}$ \( 1 + 126692 T + 464904286 p^{3} T^{2} + 126692 p^{21} T^{3} + p^{42} T^{4} \)
7$D_{4}$ \( 1 + 41799812 p T + 9706833687364722 p^{2} T^{2} + 41799812 p^{22} T^{3} + p^{42} T^{4} \)
11$D_{4}$ \( 1 + 41326831776 T + \)\(36\!\cdots\!06\)\( p T^{2} + 41326831776 p^{21} T^{3} + p^{42} T^{4} \)
13$D_{4}$ \( 1 - 145157805556 p T + \)\(81\!\cdots\!38\)\( p^{2} T^{2} - 145157805556 p^{22} T^{3} + p^{42} T^{4} \)
17$D_{4}$ \( 1 + 3100413932364 T + \)\(22\!\cdots\!74\)\( p T^{2} + 3100413932364 p^{21} T^{3} + p^{42} T^{4} \)
19$D_{4}$ \( 1 - 377673402760 p T + \)\(36\!\cdots\!58\)\( p^{2} T^{2} - 377673402760 p^{22} T^{3} + p^{42} T^{4} \)
23$D_{4}$ \( 1 + 357559990100052 T + \)\(87\!\cdots\!22\)\( T^{2} + 357559990100052 p^{21} T^{3} + p^{42} T^{4} \)
29$D_{4}$ \( 1 + 2454584202185940 T + \)\(46\!\cdots\!58\)\( T^{2} + 2454584202185940 p^{21} T^{3} + p^{42} T^{4} \)
31$D_{4}$ \( 1 + 4310677151243336 T + \)\(16\!\cdots\!86\)\( T^{2} + 4310677151243336 p^{21} T^{3} + p^{42} T^{4} \)
37$D_{4}$ \( 1 + 32389201650205724 T + \)\(12\!\cdots\!18\)\( T^{2} + 32389201650205724 p^{21} T^{3} + p^{42} T^{4} \)
41$D_{4}$ \( 1 + 16298035212712116 T + \)\(71\!\cdots\!46\)\( T^{2} + 16298035212712116 p^{21} T^{3} + p^{42} T^{4} \)
43$D_{4}$ \( 1 + 161845702253479412 T + \)\(46\!\cdots\!22\)\( T^{2} + 161845702253479412 p^{21} T^{3} + p^{42} T^{4} \)
47$D_{4}$ \( 1 - 153492993930726996 T + \)\(18\!\cdots\!98\)\( T^{2} - 153492993930726996 p^{21} T^{3} + p^{42} T^{4} \)
53$D_{4}$ \( 1 - 604198357317820308 T + \)\(20\!\cdots\!22\)\( T^{2} - 604198357317820308 p^{21} T^{3} + p^{42} T^{4} \)
59$D_{4}$ \( 1 - 3110197357974672120 T + \)\(32\!\cdots\!18\)\( T^{2} - 3110197357974672120 p^{21} T^{3} + p^{42} T^{4} \)
61$D_{4}$ \( 1 + 791142209760451676 T + \)\(37\!\cdots\!66\)\( T^{2} + 791142209760451676 p^{21} T^{3} + p^{42} T^{4} \)
67$D_{4}$ \( 1 + 5111819523383561564 T + \)\(33\!\cdots\!58\)\( T^{2} + 5111819523383561564 p^{21} T^{3} + p^{42} T^{4} \)
71$D_{4}$ \( 1 + 82388741313238482456 T + \)\(30\!\cdots\!26\)\( T^{2} + 82388741313238482456 p^{21} T^{3} + p^{42} T^{4} \)
73$D_{4}$ \( 1 - 10657918464992348548 T + \)\(27\!\cdots\!22\)\( T^{2} - 10657918464992348548 p^{21} T^{3} + p^{42} T^{4} \)
79$D_{4}$ \( 1 - 28314166125451369360 T + \)\(14\!\cdots\!58\)\( T^{2} - 28314166125451369360 p^{21} T^{3} + p^{42} T^{4} \)
83$D_{4}$ \( 1 + 125635249468995804 p T + \)\(38\!\cdots\!22\)\( T^{2} + 125635249468995804 p^{22} T^{3} + p^{42} T^{4} \)
89$D_{4}$ \( 1 - \)\(47\!\cdots\!80\)\( T + \)\(23\!\cdots\!78\)\( T^{2} - \)\(47\!\cdots\!80\)\( p^{21} T^{3} + p^{42} T^{4} \)
97$D_{4}$ \( 1 - \)\(54\!\cdots\!96\)\( T + \)\(92\!\cdots\!98\)\( T^{2} - \)\(54\!\cdots\!96\)\( p^{21} T^{3} + p^{42} T^{4} \)
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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

−15.85839538305185702401666177148, −15.12359892667593229874454707570, −13.64200455617642625052055268830, −12.99776444958657970857434160675, −11.66285262007658734522626524496, −11.61865778298513851339169783129, −10.73804741737209801406969588700, −10.48634307925554845713591773938, −9.118563010103317364979725732703, −8.492003153746732251829754573077, −7.88643975889491217582953307268, −6.79751258135651795703609932208, −6.04213711772108342139157952173, −5.64939506760527435706315109686, −3.86191063965913166125107408071, −3.42967016289538218632564970545, −1.76894170409024154823913311715, −1.11642112105821876100684601202, 0, 0, 1.11642112105821876100684601202, 1.76894170409024154823913311715, 3.42967016289538218632564970545, 3.86191063965913166125107408071, 5.64939506760527435706315109686, 6.04213711772108342139157952173, 6.79751258135651795703609932208, 7.88643975889491217582953307268, 8.492003153746732251829754573077, 9.118563010103317364979725732703, 10.48634307925554845713591773938, 10.73804741737209801406969588700, 11.61865778298513851339169783129, 11.66285262007658734522626524496, 12.99776444958657970857434160675, 13.64200455617642625052055268830, 15.12359892667593229874454707570, 15.85839538305185702401666177148

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