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Properties

Label 4-10e2-1.1-c23e2-0-0
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3·2-s + 6.86e5·3-s + 1.25e7·4-s − 9.76e7·5-s − 2.81e9·6-s − 3.52e9·7-s − 3.43e10·8-s + 2.25e11·9-s + 4.00e11·10-s + 9.36e11·11-s + 8.63e12·12-s + 9.96e12·13-s + 1.44e13·14-s − 6.70e13·15-s + 8.79e13·16-s − 1.60e12·17-s − 9.25e14·18-s − 2.56e14·19-s − 1.22e15·20-s − 2.42e15·21-s − 3.83e15·22-s + 3.59e15·23-s − 2.35e16·24-s + 7.15e15·25-s − 4.08e16·26-s + 5.13e16·27-s − 4.44e16·28-s + ⋯
L(s)  = 1  − 1.41·2-s + 2.23·3-s + 3/2·4-s − 0.894·5-s − 3.16·6-s − 0.674·7-s − 1.41·8-s + 2.40·9-s + 1.26·10-s + 0.989·11-s + 3.35·12-s + 1.54·13-s + 0.954·14-s − 2.00·15-s + 5/4·16-s − 0.0113·17-s − 3.39·18-s − 0.505·19-s − 1.34·20-s − 1.50·21-s − 1.39·22-s + 0.786·23-s − 3.16·24-s + 3/5·25-s − 2.18·26-s + 1.77·27-s − 1.01·28-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: \(0\)
Selberg data: \((4,\ 100,\ (\ :23/2, 23/2),\ 1)\)

Particular Values

\(L(12)\) \(\approx\) \(4.041866136\)
\(L(\frac12)\) \(\approx\) \(4.041866136\)
\(L(\frac{25}{2})\) 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$D_{4}$ \( 1 - 76276 p^{2} T + 112148014 p^{7} T^{2} - 76276 p^{25} T^{3} + p^{46} T^{4} \)
7$D_{4}$ \( 1 + 3529595108 T + 210532244602639698 p^{2} T^{2} + 3529595108 p^{23} T^{3} + p^{46} T^{4} \)
11$D_{4}$ \( 1 - 85141569984 p T + \)\(11\!\cdots\!86\)\( p^{2} T^{2} - 85141569984 p^{24} T^{3} + p^{46} T^{4} \)
13$D_{4}$ \( 1 - 9961846393684 T + \)\(89\!\cdots\!58\)\( T^{2} - 9961846393684 p^{23} T^{3} + p^{46} T^{4} \)
17$D_{4}$ \( 1 + 94688808204 p T + \)\(81\!\cdots\!38\)\( p^{2} T^{2} + 94688808204 p^{24} T^{3} + p^{46} T^{4} \)
19$D_{4}$ \( 1 + 256914240498200 T + \)\(19\!\cdots\!22\)\( p T^{2} + 256914240498200 p^{23} T^{3} + p^{46} T^{4} \)
23$D_{4}$ \( 1 - 3594102326172564 T + \)\(40\!\cdots\!58\)\( T^{2} - 3594102326172564 p^{23} T^{3} + p^{46} T^{4} \)
29$D_{4}$ \( 1 - 195470005256002620 T + \)\(17\!\cdots\!78\)\( T^{2} - 195470005256002620 p^{23} T^{3} + p^{46} T^{4} \)
31$D_{4}$ \( 1 - 168811393572212104 T + \)\(47\!\cdots\!86\)\( T^{2} - 168811393572212104 p^{23} T^{3} + p^{46} T^{4} \)
37$D_{4}$ \( 1 - 914879862377868532 T + \)\(91\!\cdots\!62\)\( T^{2} - 914879862377868532 p^{23} T^{3} + p^{46} T^{4} \)
41$D_{4}$ \( 1 - 1944225537689591724 T + \)\(19\!\cdots\!86\)\( T^{2} - 1944225537689591724 p^{23} T^{3} + p^{46} T^{4} \)
43$D_{4}$ \( 1 - 2162030450882223364 T + \)\(65\!\cdots\!38\)\( T^{2} - 2162030450882223364 p^{23} T^{3} + p^{46} T^{4} \)
47$D_{4}$ \( 1 - 20899427956391479692 T + \)\(33\!\cdots\!62\)\( T^{2} - 20899427956391479692 p^{23} T^{3} + p^{46} T^{4} \)
53$D_{4}$ \( 1 - 72675644730523142244 T + \)\(73\!\cdots\!38\)\( T^{2} - 72675644730523142244 p^{23} T^{3} + p^{46} T^{4} \)
59$D_{4}$ \( 1 + \)\(26\!\cdots\!60\)\( T + \)\(12\!\cdots\!58\)\( T^{2} + \)\(26\!\cdots\!60\)\( p^{23} T^{3} + p^{46} T^{4} \)
61$D_{4}$ \( 1 + \)\(34\!\cdots\!96\)\( T + \)\(25\!\cdots\!66\)\( T^{2} + \)\(34\!\cdots\!96\)\( p^{23} T^{3} + p^{46} T^{4} \)
67$D_{4}$ \( 1 - \)\(17\!\cdots\!52\)\( T + \)\(21\!\cdots\!02\)\( T^{2} - \)\(17\!\cdots\!52\)\( p^{23} T^{3} + p^{46} T^{4} \)
71$D_{4}$ \( 1 - \)\(57\!\cdots\!64\)\( T + \)\(15\!\cdots\!46\)\( T^{2} - \)\(57\!\cdots\!64\)\( p^{23} T^{3} + p^{46} T^{4} \)
73$D_{4}$ \( 1 + \)\(10\!\cdots\!76\)\( T + \)\(12\!\cdots\!78\)\( T^{2} + \)\(10\!\cdots\!76\)\( p^{23} T^{3} + p^{46} T^{4} \)
79$D_{4}$ \( 1 + \)\(51\!\cdots\!20\)\( T + \)\(56\!\cdots\!78\)\( T^{2} + \)\(51\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
83$D_{4}$ \( 1 - \)\(87\!\cdots\!44\)\( T + \)\(28\!\cdots\!58\)\( T^{2} - \)\(87\!\cdots\!44\)\( p^{23} T^{3} + p^{46} T^{4} \)
89$D_{4}$ \( 1 - \)\(24\!\cdots\!20\)\( T + \)\(14\!\cdots\!38\)\( T^{2} - \)\(24\!\cdots\!20\)\( p^{23} T^{3} + p^{46} T^{4} \)
97$D_{4}$ \( 1 - \)\(28\!\cdots\!12\)\( T + \)\(66\!\cdots\!82\)\( T^{2} - \)\(28\!\cdots\!12\)\( p^{23} T^{3} + p^{46} T^{4} \)
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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

−15.50997745658585319492659493277, −15.19117359675760961223231492886, −14.13413588294952209331189303042, −13.83123498308352105226391530833, −12.69464495104874652409336596793, −11.93366291452725969285456019802, −10.94404979533563908472560842663, −10.21339686114719998896677824859, −9.152373569808496861084334929293, −8.993758261191315894266003781862, −8.115597605634270287552641704517, −8.101830958496358041931908930425, −6.77114373542903646842451922125, −6.41371313771799990709877614830, −4.27934146385728398294270129474, −3.56112422779799500224801165446, −2.88858060961328901916219702569, −2.40954745454556075414213239421, −1.05751238210411616523404130839, −0.863674310779455764603152532422, 0.863674310779455764603152532422, 1.05751238210411616523404130839, 2.40954745454556075414213239421, 2.88858060961328901916219702569, 3.56112422779799500224801165446, 4.27934146385728398294270129474, 6.41371313771799990709877614830, 6.77114373542903646842451922125, 8.101830958496358041931908930425, 8.115597605634270287552641704517, 8.993758261191315894266003781862, 9.152373569808496861084334929293, 10.21339686114719998896677824859, 10.94404979533563908472560842663, 11.93366291452725969285456019802, 12.69464495104874652409336596793, 13.83123498308352105226391530833, 14.13413588294952209331189303042, 15.19117359675760961223231492886, 15.50997745658585319492659493277

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