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Properties

Label 4-10e2-1.1-c27e2-0-2
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.63e4·2-s − 4.70e6·3-s + 2.01e8·4-s − 2.44e9·5-s − 7.70e10·6-s − 5.71e10·7-s + 2.19e12·8-s + 1.11e13·9-s − 4.00e13·10-s + 1.69e14·11-s − 9.46e14·12-s − 1.03e15·13-s − 9.36e14·14-s + 1.14e16·15-s + 2.25e16·16-s − 5.04e16·17-s + 1.83e17·18-s + 4.46e17·19-s − 4.91e17·20-s + 2.68e17·21-s + 2.78e18·22-s − 6.64e18·23-s − 1.03e19·24-s + 4.47e18·25-s − 1.70e19·26-s − 3.70e19·27-s − 1.15e19·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.70·3-s + 3/2·4-s − 0.894·5-s − 2.40·6-s − 0.223·7-s + 1.41·8-s + 1.46·9-s − 1.26·10-s + 1.48·11-s − 2.55·12-s − 0.951·13-s − 0.315·14-s + 1.52·15-s + 5/4·16-s − 1.23·17-s + 2.07·18-s + 2.43·19-s − 1.34·20-s + 0.379·21-s + 2.09·22-s − 2.74·23-s − 2.40·24-s + 3/5·25-s − 1.34·26-s − 1.75·27-s − 0.334·28-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$D_{4}$ \( 1 + 1567652 p T + 44989959806 p^{5} T^{2} + 1567652 p^{28} T^{3} + p^{54} T^{4} \)
7$D_{4}$ \( 1 + 8169291644 p T + 53908827334167324402 p^{4} T^{2} + 8169291644 p^{28} T^{3} + p^{54} T^{4} \)
11$D_{4}$ \( 1 - 1403730832224 p^{2} T + \)\(23\!\cdots\!26\)\( p^{2} T^{2} - 1403730832224 p^{29} T^{3} + p^{54} T^{4} \)
13$D_{4}$ \( 1 + 79930383943292 p T + \)\(87\!\cdots\!02\)\( p^{2} T^{2} + 79930383943292 p^{28} T^{3} + p^{54} T^{4} \)
17$D_{4}$ \( 1 + 50445145609767948 T + \)\(38\!\cdots\!22\)\( T^{2} + 50445145609767948 p^{27} T^{3} + p^{54} T^{4} \)
19$D_{4}$ \( 1 - 446863730094123400 T + \)\(57\!\cdots\!62\)\( p T^{2} - 446863730094123400 p^{27} T^{3} + p^{54} T^{4} \)
23$D_{4}$ \( 1 + 289013388361901172 p T + \)\(39\!\cdots\!82\)\( p^{2} T^{2} + 289013388361901172 p^{28} T^{3} + p^{54} T^{4} \)
29$D_{4}$ \( 1 + 47997399844470169380 T + \)\(25\!\cdots\!18\)\( T^{2} + 47997399844470169380 p^{27} T^{3} + p^{54} T^{4} \)
31$D_{4}$ \( 1 - 33574730315678836744 T - \)\(51\!\cdots\!94\)\( T^{2} - 33574730315678836744 p^{27} T^{3} + p^{54} T^{4} \)
37$D_{4}$ \( 1 + \)\(13\!\cdots\!08\)\( T + \)\(39\!\cdots\!82\)\( T^{2} + \)\(13\!\cdots\!08\)\( p^{27} T^{3} + p^{54} T^{4} \)
41$D_{4}$ \( 1 + \)\(11\!\cdots\!56\)\( T - \)\(12\!\cdots\!54\)\( T^{2} + \)\(11\!\cdots\!56\)\( p^{27} T^{3} + p^{54} T^{4} \)
43$D_{4}$ \( 1 + \)\(39\!\cdots\!36\)\( T + \)\(13\!\cdots\!38\)\( T^{2} + \)\(39\!\cdots\!36\)\( p^{27} T^{3} + p^{54} T^{4} \)
47$D_{4}$ \( 1 + \)\(32\!\cdots\!28\)\( T + \)\(22\!\cdots\!22\)\( T^{2} + \)\(32\!\cdots\!28\)\( p^{27} T^{3} + p^{54} T^{4} \)
53$D_{4}$ \( 1 + \)\(33\!\cdots\!96\)\( T + \)\(74\!\cdots\!78\)\( T^{2} + \)\(33\!\cdots\!96\)\( p^{27} T^{3} + p^{54} T^{4} \)
59$D_{4}$ \( 1 + \)\(34\!\cdots\!60\)\( T + \)\(11\!\cdots\!38\)\( T^{2} + \)\(34\!\cdots\!60\)\( p^{27} T^{3} + p^{54} T^{4} \)
61$D_{4}$ \( 1 - \)\(22\!\cdots\!84\)\( T + \)\(21\!\cdots\!06\)\( T^{2} - \)\(22\!\cdots\!84\)\( p^{27} T^{3} + p^{54} T^{4} \)
67$D_{4}$ \( 1 + \)\(39\!\cdots\!28\)\( T + \)\(44\!\cdots\!42\)\( T^{2} + \)\(39\!\cdots\!28\)\( p^{27} T^{3} + p^{54} T^{4} \)
71$D_{4}$ \( 1 + \)\(24\!\cdots\!76\)\( T + \)\(13\!\cdots\!26\)\( T^{2} + \)\(24\!\cdots\!76\)\( p^{27} T^{3} + p^{54} T^{4} \)
73$D_{4}$ \( 1 + \)\(12\!\cdots\!96\)\( T + \)\(20\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!96\)\( p^{27} T^{3} + p^{54} T^{4} \)
79$D_{4}$ \( 1 - \)\(14\!\cdots\!80\)\( T + \)\(34\!\cdots\!18\)\( T^{2} - \)\(14\!\cdots\!80\)\( p^{27} T^{3} + p^{54} T^{4} \)
83$D_{4}$ \( 1 - \)\(11\!\cdots\!84\)\( T + \)\(15\!\cdots\!18\)\( T^{2} - \)\(11\!\cdots\!84\)\( p^{27} T^{3} + p^{54} T^{4} \)
89$D_{4}$ \( 1 - \)\(13\!\cdots\!20\)\( T + \)\(29\!\cdots\!58\)\( T^{2} - \)\(13\!\cdots\!20\)\( p^{27} T^{3} + p^{54} T^{4} \)
97$D_{4}$ \( 1 - \)\(11\!\cdots\!92\)\( T + \)\(11\!\cdots\!42\)\( T^{2} - \)\(11\!\cdots\!92\)\( p^{27} T^{3} + p^{54} T^{4} \)
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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

−14.06477204099351232904679139845, −13.38410926905362476763089848329, −12.27102297236651670439491415361, −12.07537324855380251712478265333, −11.39646639776216202051138602218, −11.37692912392080901369948708456, −10.08278243994427378294633847993, −9.429053755919024524212480593263, −7.81429847586326825084697725455, −7.27043702409289217151549939667, −6.32825728213086568247299599395, −6.12260802869509641084914037188, −5.06964209794475153978336899621, −4.70355361070169436196491332199, −3.78658085621396615232428513966, −3.39663315239867535633678402502, −1.98621832638906211321755447021, −1.32546194362356864904919110011, 0, 0, 1.32546194362356864904919110011, 1.98621832638906211321755447021, 3.39663315239867535633678402502, 3.78658085621396615232428513966, 4.70355361070169436196491332199, 5.06964209794475153978336899621, 6.12260802869509641084914037188, 6.32825728213086568247299599395, 7.27043702409289217151549939667, 7.81429847586326825084697725455, 9.429053755919024524212480593263, 10.08278243994427378294633847993, 11.37692912392080901369948708456, 11.39646639776216202051138602218, 12.07537324855380251712478265333, 12.27102297236651670439491415361, 13.38410926905362476763089848329, 14.06477204099351232904679139845

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