Properties

Label 4-1470e2-1.1-c3e2-0-2
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 − 58·11-s + 72·12-s + 56·13-s − 60·15-s + 80·16-s − 10·17-s − 108·18-s − 2·19-s − 120·20-s + 232·22-s − 130·23-s − 192·24-s + 75·25-s − 224·26-s + 108·27-s − 34·29-s + 240·30-s + 344·31-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 − 1.58·11-s + 1.73·12-s + 1.19·13-s − 1.03·15-s + 5/4·16-s − 0.142·17-s − 1.41·18-s − 0.0241·19-s − 1.34·20-s + 2.24·22-s − 1.17·23-s − 1.63·24-s + 3/5·25-s − 1.68·26-s + 0.769·27-s − 0.217·29-s + 1.46·30-s + 1.99·31-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

\(L(2)\) \(\approx\) \(2.098671387\)
\(L(\frac12)\) \(\approx\) \(2.098671387\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + 58 T + 2998 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 56 T + 3158 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 10 T - 2774 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 2 T + 1094 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 130 T + 15934 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 34 T - 12038 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 344 T + 70986 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 54 T - 43910 T^{2} + 54 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 60 T + 130662 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 206 T + 84278 T^{2} + 206 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 + 958 T + 432542 T^{2} + 958 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 1350 T + 752874 T^{2} - 1350 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 132 T + 364614 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 - 18 T + 392938 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 1486 T + 1149030 T^{2} - 1486 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 1364 T + 1130446 T^{2} + 1364 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 44 T + 382598 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 152 T + p^{3} T^{2} )^{2} \)
83$D_{4}$ \( 1 - 1892 T + 2020310 T^{2} - 1892 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 596 T + 1369462 T^{2} - 596 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 508 T + 524342 T^{2} - 508 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.032162969514075115834921888771, −8.925354584552030419995228199323, −8.339208298502336908565220608368, −8.284096041918356149189997494780, −7.87767197381752105662077587773, −7.70235764291536452490514698488, −7.12738063166916068382397149844, −6.70226964175945441585276494589, −6.30683281048248148772222244244, −5.82373154824374583238798568741, −5.14284937733468778601144976742, −4.69377180373087505831736079339, −4.07255021063233031546429247725, −3.54655468795949193398731689365, −3.18569770137285815046005262765, −2.70605574849566530714611597799, −2.06572485572992598237978005630, −1.78220339906595953224014933250, −0.72841897985124870574674577902, −0.54971134741435343764068328018, 0.54971134741435343764068328018, 0.72841897985124870574674577902, 1.78220339906595953224014933250, 2.06572485572992598237978005630, 2.70605574849566530714611597799, 3.18569770137285815046005262765, 3.54655468795949193398731689365, 4.07255021063233031546429247725, 4.69377180373087505831736079339, 5.14284937733468778601144976742, 5.82373154824374583238798568741, 6.30683281048248148772222244244, 6.70226964175945441585276494589, 7.12738063166916068382397149844, 7.70235764291536452490514698488, 7.87767197381752105662077587773, 8.284096041918356149189997494780, 8.339208298502336908565220608368, 8.925354584552030419995228199323, 9.032162969514075115834921888771

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