Properties

Label 4-1470e2-1.1-c3e2-0-3
Degree $4$
Conductor $2160900$
Sign $1$
Analytic cond. $7522.57$
Root an. cond. $9.31304$
Motivic weight $3$
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  + 4·2-s − 6·3-s + 12·4-s − 10·5-s − 24·6-s + 32·8-s + 27·9-s − 40·10-s + 32·11-s − 72·12-s + 12·13-s + 60·15-s + 80·16-s − 4·17-s + 108·18-s − 64·19-s − 120·20-s + 128·22-s + 32·23-s − 192·24-s + 75·25-s + 48·26-s − 108·27-s − 84·29-s + 240·30-s + 72·31-s + 192·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s + 1.41·8-s + 9-s − 1.26·10-s + 0.877·11-s − 1.73·12-s + 0.256·13-s + 1.03·15-s + 5/4·16-s − 0.0570·17-s + 1.41·18-s − 0.772·19-s − 1.34·20-s + 1.24·22-s + 0.290·23-s − 1.63·24-s + 3/5·25-s + 0.362·26-s − 0.769·27-s − 0.537·29-s + 1.46·30-s + 0.417·31-s + 1.06·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2160900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(7522.57\)
Root analytic conductor: \(9.31304\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2160900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.809830013\)
\(L(\frac12)\) \(\approx\) \(5.809830013\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3$C_1$ \( ( 1 + p T )^{2} \)
5$C_1$ \( ( 1 + p T )^{2} \)
7 \( 1 \)
good11$D_{4}$ \( 1 - 32 T + 1222 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 12 T - 2354 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 4 T + 8134 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 64 T - 522 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 32 T + 22894 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 84 T + 35278 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 72 T + 54094 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 60 T + 100510 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 92 T + 31414 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 184 T + 165782 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 208 T + 3746 p T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 300 T + 304990 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 568 T + 464278 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 516 T + 478126 T^{2} - 516 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 392 T + 597542 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1448 T + 1212862 T^{2} + 1448 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 756 T + 751318 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 416 T + 1022558 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 - 1752 T + 1666726 T^{2} - 1752 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 956 T + 1529878 T^{2} - 956 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 684 T + 795814 T^{2} + 684 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.308312095760455415504478218186, −8.960519624678742715666579753504, −8.421794099673240846342823758873, −8.016757496737506977353519632283, −7.39469216216241113928577805961, −7.26604751281023337146076915473, −6.70848156952837448965288614498, −6.28810172147534701338148745620, −6.16589304691016442256181253240, −5.61717487407936594178042817261, −4.97456142356607328467228436010, −4.85834516147405295198146694045, −4.27292215979894942725320433921, −3.94371970749990348771511845463, −3.52375161356166878632604775377, −3.08217761711216436483311500403, −2.16428515679126103716300865794, −1.83818106319281272416747959592, −0.862386267906064029963969317947, −0.58819448459192855493325233948, 0.58819448459192855493325233948, 0.862386267906064029963969317947, 1.83818106319281272416747959592, 2.16428515679126103716300865794, 3.08217761711216436483311500403, 3.52375161356166878632604775377, 3.94371970749990348771511845463, 4.27292215979894942725320433921, 4.85834516147405295198146694045, 4.97456142356607328467228436010, 5.61717487407936594178042817261, 6.16589304691016442256181253240, 6.28810172147534701338148745620, 6.70848156952837448965288614498, 7.26604751281023337146076915473, 7.39469216216241113928577805961, 8.016757496737506977353519632283, 8.421794099673240846342823758873, 8.960519624678742715666579753504, 9.308312095760455415504478218186

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