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Properties

Label 4-10e2-1.1-c37e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $7519.32$
Root an. cond. $9.31203$
Motivic weight $37$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.24e5·2-s − 1.85e8·3-s + 2.06e11·4-s + 7.62e12·5-s − 9.72e13·6-s − 6.51e14·7-s + 7.20e16·8-s − 5.18e17·9-s + 4.00e18·10-s − 1.58e19·11-s − 3.82e19·12-s + 6.42e18·13-s − 3.41e20·14-s − 1.41e21·15-s + 2.36e22·16-s − 3.00e22·17-s − 2.71e23·18-s − 2.82e23·19-s + 1.57e24·20-s + 1.20e23·21-s − 8.28e24·22-s − 1.43e25·23-s − 1.33e25·24-s + 4.36e25·25-s + 3.37e24·26-s + 1.15e26·27-s − 1.34e26·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.276·3-s + 3/2·4-s + 0.894·5-s − 0.390·6-s − 0.151·7-s + 1.41·8-s − 1.15·9-s + 1.26·10-s − 0.857·11-s − 0.414·12-s + 0.0158·13-s − 0.213·14-s − 0.247·15-s + 5/4·16-s − 0.518·17-s − 1.62·18-s − 0.621·19-s + 1.34·20-s + 0.0418·21-s − 1.21·22-s − 0.922·23-s − 0.390·24-s + 3/5·25-s + 0.0224·26-s + 0.381·27-s − 0.226·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+37/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(7519.32\)
Root analytic conductor: \(9.31203\)
Motivic weight: \(37\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 100,\ (\ :37/2, 37/2),\ 1)\)

Particular Values

\(L(19)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{39}{2})\) 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$C_1$ \( ( 1 - p^{18} T )^{2} \)
5$C_1$ \( ( 1 - p^{18} T )^{2} \)
good3$D_{4}$ \( 1 + 6870404 p^{3} T + 9359750919958 p^{10} T^{2} + 6870404 p^{40} T^{3} + p^{74} T^{4} \)
7$D_{4}$ \( 1 + 13297556160044 p^{2} T + \)\(14\!\cdots\!14\)\( p^{5} T^{2} + 13297556160044 p^{39} T^{3} + p^{74} T^{4} \)
11$D_{4}$ \( 1 + 130670377114549776 p^{2} T + \)\(53\!\cdots\!66\)\( p^{3} T^{2} + 130670377114549776 p^{39} T^{3} + p^{74} T^{4} \)
13$D_{4}$ \( 1 - 494564960558966044 p T + \)\(17\!\cdots\!98\)\( p^{2} T^{2} - 494564960558966044 p^{38} T^{3} + p^{74} T^{4} \)
17$D_{4}$ \( 1 + \)\(17\!\cdots\!08\)\( p T + \)\(23\!\cdots\!02\)\( p^{2} T^{2} + \)\(17\!\cdots\!08\)\( p^{38} T^{3} + p^{74} T^{4} \)
19$D_{4}$ \( 1 + \)\(28\!\cdots\!00\)\( T + \)\(17\!\cdots\!62\)\( p T^{2} + \)\(28\!\cdots\!00\)\( p^{37} T^{3} + p^{74} T^{4} \)
23$D_{4}$ \( 1 + \)\(14\!\cdots\!08\)\( T + \)\(23\!\cdots\!14\)\( p T^{2} + \)\(14\!\cdots\!08\)\( p^{37} T^{3} + p^{74} T^{4} \)
29$D_{4}$ \( 1 + \)\(57\!\cdots\!20\)\( T + \)\(49\!\cdots\!42\)\( p T^{2} + \)\(57\!\cdots\!20\)\( p^{37} T^{3} + p^{74} T^{4} \)
31$D_{4}$ \( 1 - \)\(22\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( p T^{2} - \)\(22\!\cdots\!44\)\( p^{37} T^{3} + p^{74} T^{4} \)
37$D_{4}$ \( 1 - \)\(63\!\cdots\!12\)\( p T + \)\(12\!\cdots\!18\)\( T^{2} - \)\(63\!\cdots\!12\)\( p^{38} T^{3} + p^{74} T^{4} \)
41$D_{4}$ \( 1 + \)\(33\!\cdots\!56\)\( T + \)\(58\!\cdots\!46\)\( T^{2} + \)\(33\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \)
43$D_{4}$ \( 1 + \)\(30\!\cdots\!48\)\( T + \)\(70\!\cdots\!62\)\( T^{2} + \)\(30\!\cdots\!48\)\( p^{37} T^{3} + p^{74} T^{4} \)
47$D_{4}$ \( 1 + \)\(57\!\cdots\!96\)\( T + \)\(15\!\cdots\!78\)\( T^{2} + \)\(57\!\cdots\!96\)\( p^{37} T^{3} + p^{74} T^{4} \)
53$D_{4}$ \( 1 + \)\(12\!\cdots\!28\)\( T + \)\(16\!\cdots\!22\)\( T^{2} + \)\(12\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \)
59$D_{4}$ \( 1 + \)\(47\!\cdots\!40\)\( T + \)\(66\!\cdots\!38\)\( T^{2} + \)\(47\!\cdots\!40\)\( p^{37} T^{3} + p^{74} T^{4} \)
61$D_{4}$ \( 1 - \)\(11\!\cdots\!84\)\( T + \)\(17\!\cdots\!06\)\( T^{2} - \)\(11\!\cdots\!84\)\( p^{37} T^{3} + p^{74} T^{4} \)
67$D_{4}$ \( 1 - \)\(76\!\cdots\!04\)\( T + \)\(21\!\cdots\!58\)\( T^{2} - \)\(76\!\cdots\!04\)\( p^{37} T^{3} + p^{74} T^{4} \)
71$D_{4}$ \( 1 + \)\(17\!\cdots\!76\)\( T + \)\(62\!\cdots\!26\)\( T^{2} + \)\(17\!\cdots\!76\)\( p^{37} T^{3} + p^{74} T^{4} \)
73$D_{4}$ \( 1 + \)\(25\!\cdots\!28\)\( T + \)\(18\!\cdots\!02\)\( T^{2} + \)\(25\!\cdots\!28\)\( p^{37} T^{3} + p^{74} T^{4} \)
79$D_{4}$ \( 1 + \)\(29\!\cdots\!80\)\( T + \)\(76\!\cdots\!18\)\( T^{2} + \)\(29\!\cdots\!80\)\( p^{37} T^{3} + p^{74} T^{4} \)
83$D_{4}$ \( 1 + \)\(57\!\cdots\!88\)\( T + \)\(26\!\cdots\!82\)\( T^{2} + \)\(57\!\cdots\!88\)\( p^{37} T^{3} + p^{74} T^{4} \)
89$D_{4}$ \( 1 + \)\(17\!\cdots\!20\)\( T + \)\(34\!\cdots\!58\)\( T^{2} + \)\(17\!\cdots\!20\)\( p^{37} T^{3} + p^{74} T^{4} \)
97$D_{4}$ \( 1 + \)\(24\!\cdots\!56\)\( T + \)\(91\!\cdots\!58\)\( T^{2} + \)\(24\!\cdots\!56\)\( p^{37} T^{3} + p^{74} T^{4} \)
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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

−12.73660161638334851055762928733, −12.18514768756111852234548599902, −11.24845231506530248069580810137, −11.07158561625173846818822523507, −10.10965264368049374098657355062, −9.605670757377109707356472886777, −8.364178216760477204031401219148, −8.069888683392812894331537428527, −6.71532226905801905734105790779, −6.53538917979657866022564063014, −5.65100917576693380863948649946, −5.40987349777463850717875181957, −4.68731911290903602480543382555, −3.97178720173430513063504440393, −2.91875255903165886086164931899, −2.83658219537248301860475982231, −1.90664924263849290447410384716, −1.48083451459657877538021698046, 0, 0, 1.48083451459657877538021698046, 1.90664924263849290447410384716, 2.83658219537248301860475982231, 2.91875255903165886086164931899, 3.97178720173430513063504440393, 4.68731911290903602480543382555, 5.40987349777463850717875181957, 5.65100917576693380863948649946, 6.53538917979657866022564063014, 6.71532226905801905734105790779, 8.069888683392812894331537428527, 8.364178216760477204031401219148, 9.605670757377109707356472886777, 10.10965264368049374098657355062, 11.07158561625173846818822523507, 11.24845231506530248069580810137, 12.18514768756111852234548599902, 12.73660161638334851055762928733

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