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Properties

Label 4-10e2-1.1-c33e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $4758.63$
Root an. cond. $8.30559$
Motivic weight $33$
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  + 1.31e5·2-s − 7.46e7·3-s + 1.28e10·4-s + 3.05e11·5-s − 9.78e12·6-s + 2.03e13·7-s + 1.12e15·8-s − 5.79e15·9-s + 4.00e16·10-s − 1.12e16·11-s − 9.61e17·12-s + 8.07e16·13-s + 2.66e18·14-s − 2.27e19·15-s + 9.22e19·16-s − 2.05e20·17-s − 7.59e20·18-s − 2.43e21·19-s + 3.93e21·20-s − 1.51e21·21-s − 1.47e21·22-s + 2.90e22·23-s − 8.40e22·24-s + 6.98e22·25-s + 1.05e22·26-s + 8.65e23·27-s + 2.61e23·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.00·3-s + 3/2·4-s + 0.894·5-s − 1.41·6-s + 0.231·7-s + 1.41·8-s − 1.04·9-s + 1.26·10-s − 0.0740·11-s − 1.50·12-s + 0.0336·13-s + 0.326·14-s − 0.895·15-s + 5/4·16-s − 1.02·17-s − 1.47·18-s − 1.93·19-s + 1.34·20-s − 0.231·21-s − 0.104·22-s + 0.987·23-s − 1.41·24-s + 3/5·25-s + 0.0476·26-s + 2.08·27-s + 0.346·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+33/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(4758.63\)
Root analytic conductor: \(8.30559\)
Motivic weight: \(33\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{35}{2})\) 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$C_1$ \( ( 1 - p^{16} T )^{2} \)
5$C_1$ \( ( 1 - p^{16} T )^{2} \)
good3$D_{4}$ \( 1 + 2764756 p^{3} T + 577307045954 p^{9} T^{2} + 2764756 p^{36} T^{3} + p^{66} T^{4} \)
7$D_{4}$ \( 1 - 2903969916508 p T + \)\(33\!\cdots\!98\)\( p^{4} T^{2} - 2903969916508 p^{34} T^{3} + p^{66} T^{4} \)
11$D_{4}$ \( 1 + 1025675198463216 p T + \)\(26\!\cdots\!86\)\( p^{2} T^{2} + 1025675198463216 p^{34} T^{3} + p^{66} T^{4} \)
13$D_{4}$ \( 1 - 6212367829542076 p T - \)\(20\!\cdots\!82\)\( p^{2} T^{2} - 6212367829542076 p^{34} T^{3} + p^{66} T^{4} \)
17$D_{4}$ \( 1 + \)\(20\!\cdots\!24\)\( T + \)\(82\!\cdots\!18\)\( T^{2} + \)\(20\!\cdots\!24\)\( p^{33} T^{3} + p^{66} T^{4} \)
19$D_{4}$ \( 1 + \)\(24\!\cdots\!00\)\( T + \)\(24\!\cdots\!22\)\( p T^{2} + \)\(24\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \)
23$D_{4}$ \( 1 - \)\(29\!\cdots\!48\)\( T + \)\(69\!\cdots\!54\)\( p T^{2} - \)\(29\!\cdots\!48\)\( p^{33} T^{3} + p^{66} T^{4} \)
29$D_{4}$ \( 1 - \)\(66\!\cdots\!80\)\( T + \)\(12\!\cdots\!82\)\( p T^{2} - \)\(66\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
31$D_{4}$ \( 1 - \)\(33\!\cdots\!04\)\( T + \)\(43\!\cdots\!06\)\( p T^{2} - \)\(33\!\cdots\!04\)\( p^{33} T^{3} + p^{66} T^{4} \)
37$D_{4}$ \( 1 + \)\(18\!\cdots\!24\)\( T + \)\(17\!\cdots\!38\)\( T^{2} + \)\(18\!\cdots\!24\)\( p^{33} T^{3} + p^{66} T^{4} \)
41$D_{4}$ \( 1 - \)\(14\!\cdots\!24\)\( T + \)\(14\!\cdots\!86\)\( T^{2} - \)\(14\!\cdots\!24\)\( p^{33} T^{3} + p^{66} T^{4} \)
43$D_{4}$ \( 1 + \)\(60\!\cdots\!52\)\( T + \)\(88\!\cdots\!62\)\( T^{2} + \)\(60\!\cdots\!52\)\( p^{33} T^{3} + p^{66} T^{4} \)
47$D_{4}$ \( 1 + \)\(66\!\cdots\!44\)\( T + \)\(41\!\cdots\!38\)\( T^{2} + \)\(66\!\cdots\!44\)\( p^{33} T^{3} + p^{66} T^{4} \)
53$D_{4}$ \( 1 + \)\(23\!\cdots\!92\)\( T - \)\(18\!\cdots\!38\)\( T^{2} + \)\(23\!\cdots\!92\)\( p^{33} T^{3} + p^{66} T^{4} \)
59$D_{4}$ \( 1 - \)\(22\!\cdots\!60\)\( T + \)\(16\!\cdots\!58\)\( T^{2} - \)\(22\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \)
61$D_{4}$ \( 1 + \)\(33\!\cdots\!96\)\( T + \)\(19\!\cdots\!66\)\( T^{2} + \)\(33\!\cdots\!96\)\( p^{33} T^{3} + p^{66} T^{4} \)
67$D_{4}$ \( 1 + \)\(25\!\cdots\!64\)\( T + \)\(44\!\cdots\!98\)\( T^{2} + \)\(25\!\cdots\!64\)\( p^{33} T^{3} + p^{66} T^{4} \)
71$D_{4}$ \( 1 + \)\(87\!\cdots\!36\)\( T + \)\(42\!\cdots\!46\)\( T^{2} + \)\(87\!\cdots\!36\)\( p^{33} T^{3} + p^{66} T^{4} \)
73$D_{4}$ \( 1 + \)\(80\!\cdots\!32\)\( T + \)\(66\!\cdots\!22\)\( T^{2} + \)\(80\!\cdots\!32\)\( p^{33} T^{3} + p^{66} T^{4} \)
79$D_{4}$ \( 1 + \)\(19\!\cdots\!80\)\( T + \)\(65\!\cdots\!78\)\( T^{2} + \)\(19\!\cdots\!80\)\( p^{33} T^{3} + p^{66} T^{4} \)
83$D_{4}$ \( 1 + \)\(27\!\cdots\!92\)\( T + \)\(25\!\cdots\!42\)\( T^{2} + \)\(27\!\cdots\!92\)\( p^{33} T^{3} + p^{66} T^{4} \)
89$D_{4}$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(28\!\cdots\!38\)\( T^{2} + \)\(15\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \)
97$D_{4}$ \( 1 + \)\(61\!\cdots\!84\)\( T + \)\(62\!\cdots\!18\)\( T^{2} + \)\(61\!\cdots\!84\)\( p^{33} T^{3} + p^{66} T^{4} \)
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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

−12.89383625080987991795024700497, −12.85118011368822357487856761933, −11.58192177592124405692874893732, −11.55596914031915609647500578712, −10.52376932753410555792810757421, −10.42347198496466689586768067043, −8.800970544287820777189424477440, −8.573097156331468292096075667245, −7.17559848908568966657586865605, −6.35489110182631321256905923617, −6.23422965452623027649869639468, −5.48695693138631109521790328039, −4.76759483324653587227046770781, −4.54955443710652652625236884855, −3.18021190696223331447625657936, −2.85631086388946562924965818818, −1.88134754992078329305752093878, −1.49525265437449679102209993794, 0, 0, 1.49525265437449679102209993794, 1.88134754992078329305752093878, 2.85631086388946562924965818818, 3.18021190696223331447625657936, 4.54955443710652652625236884855, 4.76759483324653587227046770781, 5.48695693138631109521790328039, 6.23422965452623027649869639468, 6.35489110182631321256905923617, 7.17559848908568966657586865605, 8.573097156331468292096075667245, 8.800970544287820777189424477440, 10.42347198496466689586768067043, 10.52376932753410555792810757421, 11.55596914031915609647500578712, 11.58192177592124405692874893732, 12.85118011368822357487856761933, 12.89383625080987991795024700497

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