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Properties

Label 4-90e2-1.1-c19e2-0-1
Degree $4$
Conductor $8100$
Sign $1$
Analytic cond. $42409.2$
Root an. cond. $14.3504$
Motivic weight $19$
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  + 1.02e3·2-s + 7.86e5·4-s − 3.90e6·5-s + 2.20e7·7-s + 5.36e8·8-s − 4.00e9·10-s − 1.79e9·11-s − 3.19e10·13-s + 2.25e10·14-s + 3.43e11·16-s + 7.83e11·17-s − 1.19e12·19-s − 3.07e12·20-s − 1.83e12·22-s − 6.87e12·23-s + 1.14e13·25-s − 3.27e13·26-s + 1.73e13·28-s − 1.07e14·29-s + 2.22e14·31-s + 2.11e14·32-s + 8.02e14·34-s − 8.60e13·35-s + 1.10e15·37-s − 1.22e15·38-s − 2.09e15·40-s − 8.34e14·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.206·7-s + 1.41·8-s − 1.26·10-s − 0.229·11-s − 0.835·13-s + 0.291·14-s + 5/4·16-s + 1.60·17-s − 0.851·19-s − 1.34·20-s − 0.324·22-s − 0.795·23-s + 3/5·25-s − 1.18·26-s + 0.309·28-s − 1.37·29-s + 1.51·31-s + 1.06·32-s + 2.26·34-s − 0.184·35-s + 1.39·37-s − 1.20·38-s − 1.26·40-s − 0.398·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(42409.2\)
Root analytic conductor: \(14.3504\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8100,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(\approx\) \(6.590012147\)
\(L(\frac12)\) \(\approx\) \(6.590012147\)
\(L(\frac{21}{2})\) 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$C_1$ \( ( 1 - p^{9} T )^{2} \)
3 \( 1 \)
5$C_1$ \( ( 1 + p^{9} T )^{2} \)
good7$D_{4}$ \( 1 - 22039888 T + 1092649796386146 p T^{2} - 22039888 p^{19} T^{3} + p^{38} T^{4} \)
11$D_{4}$ \( 1 + 1795098984 T + 629636543815505186 p T^{2} + 1795098984 p^{19} T^{3} + p^{38} T^{4} \)
13$D_{4}$ \( 1 + 31942891844 T + \)\(24\!\cdots\!26\)\( p T^{2} + 31942891844 p^{19} T^{3} + p^{38} T^{4} \)
17$D_{4}$ \( 1 - 46104639276 p T + \)\(19\!\cdots\!98\)\( p^{2} T^{2} - 46104639276 p^{20} T^{3} + p^{38} T^{4} \)
19$D_{4}$ \( 1 + 1197087422840 T + \)\(32\!\cdots\!58\)\( T^{2} + 1197087422840 p^{19} T^{3} + p^{38} T^{4} \)
23$D_{4}$ \( 1 + 6873503008176 T + \)\(10\!\cdots\!18\)\( T^{2} + 6873503008176 p^{19} T^{3} + p^{38} T^{4} \)
29$D_{4}$ \( 1 + 107348434878540 T + \)\(11\!\cdots\!38\)\( T^{2} + 107348434878540 p^{19} T^{3} + p^{38} T^{4} \)
31$D_{4}$ \( 1 - 222374584957744 T + \)\(53\!\cdots\!26\)\( T^{2} - 222374584957744 p^{19} T^{3} + p^{38} T^{4} \)
37$D_{4}$ \( 1 - 1105316119039948 T + \)\(13\!\cdots\!22\)\( T^{2} - 1105316119039948 p^{19} T^{3} + p^{38} T^{4} \)
41$D_{4}$ \( 1 + 834855144051924 T + \)\(43\!\cdots\!66\)\( T^{2} + 834855144051924 p^{19} T^{3} + p^{38} T^{4} \)
43$D_{4}$ \( 1 + 2383285026281384 T + \)\(21\!\cdots\!78\)\( T^{2} + 2383285026281384 p^{19} T^{3} + p^{38} T^{4} \)
47$D_{4}$ \( 1 - 621619024021632 T + \)\(26\!\cdots\!22\)\( T^{2} - 621619024021632 p^{19} T^{3} + p^{38} T^{4} \)
53$D_{4}$ \( 1 + 5531387869738236 T + \)\(10\!\cdots\!58\)\( T^{2} + 5531387869738236 p^{19} T^{3} + p^{38} T^{4} \)
59$D_{4}$ \( 1 + 15552755068197480 T + \)\(29\!\cdots\!78\)\( T^{2} + 15552755068197480 p^{19} T^{3} + p^{38} T^{4} \)
61$D_{4}$ \( 1 + 44430818356154516 T + \)\(10\!\cdots\!46\)\( T^{2} + 44430818356154516 p^{19} T^{3} + p^{38} T^{4} \)
67$D_{4}$ \( 1 + 113122907010764792 T + \)\(96\!\cdots\!22\)\( T^{2} + 113122907010764792 p^{19} T^{3} + p^{38} T^{4} \)
71$D_{4}$ \( 1 - 1164675980879661936 T + \)\(62\!\cdots\!86\)\( T^{2} - 1164675980879661936 p^{19} T^{3} + p^{38} T^{4} \)
73$D_{4}$ \( 1 + 818292079460930924 T + \)\(67\!\cdots\!18\)\( T^{2} + 818292079460930924 p^{19} T^{3} + p^{38} T^{4} \)
79$D_{4}$ \( 1 + 536821059699125360 T + \)\(23\!\cdots\!38\)\( T^{2} + 536821059699125360 p^{19} T^{3} + p^{38} T^{4} \)
83$D_{4}$ \( 1 - 719201380581658104 T + \)\(19\!\cdots\!98\)\( T^{2} - 719201380581658104 p^{19} T^{3} + p^{38} T^{4} \)
89$D_{4}$ \( 1 - 3510814279969627980 T + \)\(17\!\cdots\!18\)\( T^{2} - 3510814279969627980 p^{19} T^{3} + p^{38} T^{4} \)
97$D_{4}$ \( 1 - 5354850116637209668 T + \)\(77\!\cdots\!22\)\( T^{2} - 5354850116637209668 p^{19} T^{3} + p^{38} T^{4} \)
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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

−11.01714113964780393724388056506, −10.52845482881095466239955970851, −9.854563612788266585004543985476, −9.551309435281897043717324389469, −8.279608602709556927432904231336, −8.218965799304651294066463136980, −7.37469491105062508426885457115, −7.35403883810434587449349785849, −6.23776227453332074835610867470, −6.15593939468020146830522777750, −5.23588786365601467331786256544, −4.96272778978856319177767288111, −4.25449777384091724290134197003, −4.00310037866641388322089033738, −3.11684711162268566932872611931, −3.06582069291253520042918605342, −2.12627745981484019061834171767, −1.77758019260246656031124817610, −0.895839559009905349628516945238, −0.39868515562687079964634275102, 0.39868515562687079964634275102, 0.895839559009905349628516945238, 1.77758019260246656031124817610, 2.12627745981484019061834171767, 3.06582069291253520042918605342, 3.11684711162268566932872611931, 4.00310037866641388322089033738, 4.25449777384091724290134197003, 4.96272778978856319177767288111, 5.23588786365601467331786256544, 6.15593939468020146830522777750, 6.23776227453332074835610867470, 7.35403883810434587449349785849, 7.37469491105062508426885457115, 8.218965799304651294066463136980, 8.279608602709556927432904231336, 9.551309435281897043717324389469, 9.854563612788266585004543985476, 10.52845482881095466239955970851, 11.01714113964780393724388056506

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