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Properties

Label 4-10e2-1.1-c29e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $2838.54$
Root an. cond. $7.29918$
Motivic weight $29$
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  + 3.27e4·2-s − 3.64e6·3-s + 8.05e8·4-s + 1.22e10·5-s − 1.19e11·6-s − 2.61e12·7-s + 1.75e13·8-s − 3.69e13·9-s + 4.00e14·10-s − 6.09e14·11-s − 2.93e15·12-s − 2.30e16·13-s − 8.58e16·14-s − 4.44e16·15-s + 3.60e17·16-s − 2.12e17·17-s − 1.21e18·18-s − 1.55e18·19-s + 9.83e18·20-s + 9.54e18·21-s − 1.99e19·22-s + 4.85e19·23-s − 6.41e19·24-s + 1.11e20·25-s − 7.56e20·26-s + 6.75e19·27-s − 2.10e21·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.439·3-s + 3/2·4-s + 0.894·5-s − 0.622·6-s − 1.46·7-s + 1.41·8-s − 0.538·9-s + 1.26·10-s − 0.484·11-s − 0.659·12-s − 1.62·13-s − 2.06·14-s − 0.393·15-s + 5/4·16-s − 0.306·17-s − 0.761·18-s − 0.445·19-s + 1.34·20-s + 0.642·21-s − 0.684·22-s + 0.873·23-s − 0.622·24-s + 3/5·25-s − 2.29·26-s + 0.118·27-s − 2.19·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+29/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(2838.54\)
Root analytic conductor: \(7.29918\)
Motivic weight: \(29\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :29/2, 29/2),\ 1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) 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$C_1$ \( ( 1 - p^{14} T )^{2} \)
5$C_1$ \( ( 1 - p^{14} T )^{2} \)
good3$D_{4}$ \( 1 + 404972 p^{2} T + 22964109266 p^{7} T^{2} + 404972 p^{31} T^{3} + p^{58} T^{4} \)
7$D_{4}$ \( 1 + 374284647188 p T + \)\(23\!\cdots\!46\)\( p^{3} T^{2} + 374284647188 p^{30} T^{3} + p^{58} T^{4} \)
11$D_{4}$ \( 1 + 5038027988496 p^{2} T + \)\(13\!\cdots\!66\)\( p^{3} T^{2} + 5038027988496 p^{31} T^{3} + p^{58} T^{4} \)
13$D_{4}$ \( 1 + 1775541995838116 p T + \)\(28\!\cdots\!98\)\( p^{2} T^{2} + 1775541995838116 p^{30} T^{3} + p^{58} T^{4} \)
17$D_{4}$ \( 1 + 212788505698107756 T + \)\(45\!\cdots\!34\)\( p T^{2} + 212788505698107756 p^{29} T^{3} + p^{58} T^{4} \)
19$D_{4}$ \( 1 + 1552916806248440360 T + \)\(87\!\cdots\!82\)\( p T^{2} + 1552916806248440360 p^{29} T^{3} + p^{58} T^{4} \)
23$D_{4}$ \( 1 - 91813257293395308 p^{2} T + \)\(12\!\cdots\!58\)\( p^{2} T^{2} - 91813257293395308 p^{31} T^{3} + p^{58} T^{4} \)
29$D_{4}$ \( 1 + \)\(95\!\cdots\!40\)\( T + \)\(14\!\cdots\!38\)\( T^{2} + \)\(95\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
31$D_{4}$ \( 1 + \)\(65\!\cdots\!56\)\( T + \)\(34\!\cdots\!26\)\( T^{2} + \)\(65\!\cdots\!56\)\( p^{29} T^{3} + p^{58} T^{4} \)
37$D_{4}$ \( 1 - \)\(66\!\cdots\!64\)\( T + \)\(70\!\cdots\!78\)\( T^{2} - \)\(66\!\cdots\!64\)\( p^{29} T^{3} + p^{58} T^{4} \)
41$D_{4}$ \( 1 + \)\(91\!\cdots\!76\)\( T + \)\(91\!\cdots\!66\)\( T^{2} + \)\(91\!\cdots\!76\)\( p^{29} T^{3} + p^{58} T^{4} \)
43$D_{4}$ \( 1 + \)\(23\!\cdots\!88\)\( T + \)\(11\!\cdots\!22\)\( T^{2} + \)\(23\!\cdots\!88\)\( p^{29} T^{3} + p^{58} T^{4} \)
47$D_{4}$ \( 1 - \)\(31\!\cdots\!24\)\( T + \)\(61\!\cdots\!78\)\( T^{2} - \)\(31\!\cdots\!24\)\( p^{29} T^{3} + p^{58} T^{4} \)
53$D_{4}$ \( 1 + \)\(20\!\cdots\!48\)\( T + \)\(27\!\cdots\!42\)\( T^{2} + \)\(20\!\cdots\!48\)\( p^{29} T^{3} + p^{58} T^{4} \)
59$D_{4}$ \( 1 + \)\(56\!\cdots\!80\)\( T + \)\(37\!\cdots\!78\)\( T^{2} + \)\(56\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
61$D_{4}$ \( 1 + \)\(19\!\cdots\!16\)\( T + \)\(18\!\cdots\!46\)\( T^{2} + \)\(19\!\cdots\!16\)\( p^{29} T^{3} + p^{58} T^{4} \)
67$D_{4}$ \( 1 + \)\(19\!\cdots\!56\)\( T + \)\(14\!\cdots\!78\)\( T^{2} + \)\(19\!\cdots\!56\)\( p^{29} T^{3} + p^{58} T^{4} \)
71$D_{4}$ \( 1 - \)\(88\!\cdots\!64\)\( T + \)\(97\!\cdots\!86\)\( T^{2} - \)\(88\!\cdots\!64\)\( p^{29} T^{3} + p^{58} T^{4} \)
73$D_{4}$ \( 1 + \)\(12\!\cdots\!68\)\( T + \)\(16\!\cdots\!82\)\( T^{2} + \)\(12\!\cdots\!68\)\( p^{29} T^{3} + p^{58} T^{4} \)
79$D_{4}$ \( 1 + \)\(55\!\cdots\!40\)\( T + \)\(21\!\cdots\!38\)\( T^{2} + \)\(55\!\cdots\!40\)\( p^{29} T^{3} + p^{58} T^{4} \)
83$D_{4}$ \( 1 - \)\(14\!\cdots\!72\)\( T + \)\(13\!\cdots\!02\)\( T^{2} - \)\(14\!\cdots\!72\)\( p^{29} T^{3} + p^{58} T^{4} \)
89$D_{4}$ \( 1 - \)\(37\!\cdots\!80\)\( T + \)\(83\!\cdots\!18\)\( T^{2} - \)\(37\!\cdots\!80\)\( p^{29} T^{3} + p^{58} T^{4} \)
97$D_{4}$ \( 1 + \)\(23\!\cdots\!76\)\( T + \)\(25\!\cdots\!78\)\( T^{2} + \)\(23\!\cdots\!76\)\( p^{29} T^{3} + p^{58} T^{4} \)
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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

−13.47122809353914777495695037298, −13.17600699872056750082048963654, −12.54520928142389245067565693040, −12.07521850917296614671572280839, −10.99402256385764036356331532487, −10.57132827178239129074568349289, −9.572656168637783902356161680860, −9.204904545039867475866760619801, −7.64516201288252854182410150876, −7.05379799300279339513478686244, −6.12825567625343865066301396521, −5.97799372651180405919884713195, −5.04523612758423526662815307696, −4.59964177228313023993288801237, −3.39772588678431524498655570298, −2.88650740955351549416613284078, −2.32884995884816183750435443939, −1.50265197202637292638843371068, 0, 0, 1.50265197202637292638843371068, 2.32884995884816183750435443939, 2.88650740955351549416613284078, 3.39772588678431524498655570298, 4.59964177228313023993288801237, 5.04523612758423526662815307696, 5.97799372651180405919884713195, 6.12825567625343865066301396521, 7.05379799300279339513478686244, 7.64516201288252854182410150876, 9.204904545039867475866760619801, 9.572656168637783902356161680860, 10.57132827178239129074568349289, 10.99402256385764036356331532487, 12.07521850917296614671572280839, 12.54520928142389245067565693040, 13.17600699872056750082048963654, 13.47122809353914777495695037298

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