Properties

Label 4-9360e2-1.1-c1e2-0-3
Degree $4$
Conductor $87609600$
Sign $1$
Analytic cond. $5586.06$
Root an. cond. $8.64522$
Motivic weight $1$
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  − 2·5-s + 4·7-s − 2·13-s + 4·17-s + 8·19-s − 4·23-s + 3·25-s + 8·29-s − 8·35-s − 8·37-s − 4·41-s + 12·43-s + 4·47-s − 2·49-s + 12·53-s − 8·59-s + 8·61-s + 4·65-s + 4·67-s + 8·73-s − 8·79-s − 4·83-s − 8·85-s + 20·89-s − 8·91-s − 16·95-s + 12·97-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.970·17-s + 1.83·19-s − 0.834·23-s + 3/5·25-s + 1.48·29-s − 1.35·35-s − 1.31·37-s − 0.624·41-s + 1.82·43-s + 0.583·47-s − 2/7·49-s + 1.64·53-s − 1.04·59-s + 1.02·61-s + 0.496·65-s + 0.488·67-s + 0.936·73-s − 0.900·79-s − 0.439·83-s − 0.867·85-s + 2.11·89-s − 0.838·91-s − 1.64·95-s + 1.21·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(87609600\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(5586.06\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 87609600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.684945662\)
\(L(\frac12)\) \(\approx\) \(3.684945662\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_1$ \( ( 1 + T )^{2} \)
13$C_1$ \( ( 1 + T )^{2} \)
good7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 36 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 60 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 10 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 154 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 166 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−7.73062124043240285462684451222, −7.67278788218224011395944519119, −7.24423508388957086381107705969, −7.14086898888187674796335854032, −6.43227660923018124315522370434, −6.33338901691194358094908696144, −5.53831848251073328039271571195, −5.47177264929623354711064192585, −5.05091442003171140767331362424, −4.92016261095170395257687119079, −4.28268306770360840334649821284, −4.18404495540452294242250375167, −3.56498227402255761940102059280, −3.36029968983338503492103612149, −2.83369758938226059967890580707, −2.44256752198282896226369637712, −1.91935393646919881881982254028, −1.39117088975555908498420080548, −0.975905447413088979062451502457, −0.51717663683854937571464565733, 0.51717663683854937571464565733, 0.975905447413088979062451502457, 1.39117088975555908498420080548, 1.91935393646919881881982254028, 2.44256752198282896226369637712, 2.83369758938226059967890580707, 3.36029968983338503492103612149, 3.56498227402255761940102059280, 4.18404495540452294242250375167, 4.28268306770360840334649821284, 4.92016261095170395257687119079, 5.05091442003171140767331362424, 5.47177264929623354711064192585, 5.53831848251073328039271571195, 6.33338901691194358094908696144, 6.43227660923018124315522370434, 7.14086898888187674796335854032, 7.24423508388957086381107705969, 7.67278788218224011395944519119, 7.73062124043240285462684451222

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