Properties

Label 4-90e2-1.1-c5e2-0-1
Degree $4$
Conductor $8100$
Sign $1$
Analytic cond. $208.356$
Root an. cond. $3.79928$
Motivic weight $5$
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  − 16·4-s + 110·5-s + 1.00e3·11-s + 256·16-s + 2.68e3·19-s − 1.76e3·20-s + 8.97e3·25-s − 5.29e3·29-s − 1.12e4·31-s + 3.79e4·41-s − 1.60e4·44-s + 3.35e4·49-s + 1.10e5·55-s − 5.66e4·59-s + 3.65e4·61-s − 4.09e3·64-s + 5.76e4·71-s − 4.30e4·76-s − 1.20e5·79-s + 2.81e4·80-s + 4.53e4·89-s + 2.95e5·95-s − 1.43e5·100-s − 3.35e5·101-s + 1.07e5·109-s + 8.46e4·116-s + 4.27e5·121-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.96·5-s + 2.49·11-s + 1/4·16-s + 1.70·19-s − 0.983·20-s + 2.87·25-s − 1.16·29-s − 2.09·31-s + 3.52·41-s − 1.24·44-s + 1.99·49-s + 4.90·55-s − 2.11·59-s + 1.25·61-s − 1/8·64-s + 1.35·71-s − 0.854·76-s − 2.17·79-s + 0.491·80-s + 0.606·89-s + 3.36·95-s − 1.43·100-s − 3.27·101-s + 0.867·109-s + 0.584·116-s + 2.65·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8100\)    =    \(2^{2} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(208.356\)
Root analytic conductor: \(3.79928\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{90} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8100,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.522807585\)
\(L(\frac12)\) \(\approx\) \(4.522807585\)
\(L(\frac{7}{2})\) 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$C_2$ \( 1 + p^{4} T^{2} \)
3 \( 1 \)
5$C_2$ \( 1 - 22 p T + p^{5} T^{2} \)
good7$C_2^2$ \( 1 - 33598 T^{2} + p^{10} T^{4} \)
11$C_2$ \( ( 1 - 500 T + p^{5} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 659642 T^{2} + p^{10} T^{4} \)
17$C_2^2$ \( 1 - 541458 T^{2} + p^{10} T^{4} \)
19$C_2$ \( ( 1 - 1344 T + p^{5} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 3937314 T^{2} + p^{10} T^{4} \)
29$C_2$ \( ( 1 + 2646 T + p^{5} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 5612 T + p^{5} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 85572970 T^{2} + p^{10} T^{4} \)
41$C_2$ \( ( 1 - 18986 T + p^{5} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 288237670 T^{2} + p^{10} T^{4} \)
47$C_2^2$ \( 1 - 379480014 T^{2} + p^{10} T^{4} \)
53$C_2^2$ \( 1 + 747967430 T^{2} + p^{10} T^{4} \)
59$C_2$ \( ( 1 + 28300 T + p^{5} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 18290 T + p^{5} T^{2} )^{2} \)
67$C_2^2$ \( 1 + 1649943722 T^{2} + p^{10} T^{4} \)
71$C_2$ \( ( 1 - 28800 T + p^{5} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3197010322 T^{2} + p^{10} T^{4} \)
79$C_2$ \( ( 1 + 60228 T + p^{5} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 7871990262 T^{2} + p^{10} T^{4} \)
89$C_2$ \( ( 1 - 22678 T + p^{5} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 15808047490 T^{2} + p^{10} T^{4} \)
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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

−13.59008462153963374369562770331, −12.81870899589157661423806784592, −12.59253517828722496480906700930, −11.85336586988649920030853644940, −11.15086976163081384741291349722, −10.77556339175268371557499246693, −9.800657460147906016716225429453, −9.477751498930544868759119109792, −9.040172510451602001855129483472, −9.005040534915891400101713237691, −7.58352100319190299151170309654, −7.08438496006970519398732240640, −6.32560384392382691006213829588, −5.68668026225452014553878987905, −5.43232744380374293338938650887, −4.25716412788135121132275490878, −3.64115891137828773068691009250, −2.50475634595175616871834590523, −1.48304210697544942514375858582, −1.02735312446622005614696693660, 1.02735312446622005614696693660, 1.48304210697544942514375858582, 2.50475634595175616871834590523, 3.64115891137828773068691009250, 4.25716412788135121132275490878, 5.43232744380374293338938650887, 5.68668026225452014553878987905, 6.32560384392382691006213829588, 7.08438496006970519398732240640, 7.58352100319190299151170309654, 9.005040534915891400101713237691, 9.040172510451602001855129483472, 9.477751498930544868759119109792, 9.800657460147906016716225429453, 10.77556339175268371557499246693, 11.15086976163081384741291349722, 11.85336586988649920030853644940, 12.59253517828722496480906700930, 12.81870899589157661423806784592, 13.59008462153963374369562770331

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