Properties

Label 4-1470e2-1.1-c3e2-0-9
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 $2$

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 − 16·11-s + 72·12-s + 28·13-s − 60·15-s + 80·16-s + 32·17-s − 108·18-s − 100·19-s − 120·20-s + 64·22-s − 4·23-s − 192·24-s + 75·25-s − 112·26-s + 108·27-s + 148·29-s + 240·30-s − 132·31-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 − 0.438·11-s + 1.73·12-s + 0.597·13-s − 1.03·15-s + 5/4·16-s + 0.456·17-s − 1.41·18-s − 1.20·19-s − 1.34·20-s + 0.620·22-s − 0.0362·23-s − 1.63·24-s + 3/5·25-s − 0.844·26-s + 0.769·27-s + 0.947·29-s + 1.46·30-s − 0.764·31-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: \(2\)
Selberg data: \((4,\ 2160900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 + 16 T + 2214 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 - 28 T + 4390 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 32 T + 10050 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 100 T + 14296 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 13680 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 148 T + 53454 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 132 T + 63216 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 + 460 T + 149206 T^{2} + 460 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 284 T + 76398 T^{2} - 284 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 192 T + 168222 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 316 T + 224922 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 235112 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 344 T + 358734 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 416 T + 419608 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 184 T + 574078 T^{2} - 184 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 64 T + 704678 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 1052 T + 809710 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 668 T + 1087834 T^{2} + 668 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 68 T + 840530 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 + 468 T + 1318894 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 444 T + 620102 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \)
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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

−8.779785897911167683040466791550, −8.479042014239763599117093972148, −8.321849542673526551939410982071, −7.917729052537548775644434941620, −7.44588481357580159650206021559, −7.21540277290606944356800222118, −6.57808346500261924244310345301, −6.55211077311833451801489099963, −5.55507902239688055635784684635, −5.43815176401465270082964340457, −4.43011357653329757317756362873, −4.21493838202730050009554821129, −3.52356603276467525168971168330, −3.27434620099753897524482864912, −2.51497033910384236245797463782, −2.34574950943230043540063162745, −1.33556397557428726290086073268, −1.26661667115964919608025712018, 0, 0, 1.26661667115964919608025712018, 1.33556397557428726290086073268, 2.34574950943230043540063162745, 2.51497033910384236245797463782, 3.27434620099753897524482864912, 3.52356603276467525168971168330, 4.21493838202730050009554821129, 4.43011357653329757317756362873, 5.43815176401465270082964340457, 5.55507902239688055635784684635, 6.55211077311833451801489099963, 6.57808346500261924244310345301, 7.21540277290606944356800222118, 7.44588481357580159650206021559, 7.917729052537548775644434941620, 8.321849542673526551939410982071, 8.479042014239763599117093972148, 8.779785897911167683040466791550

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