Properties

Label 4-210e2-1.1-c7e2-0-3
Degree $4$
Conductor $44100$
Sign $1$
Analytic cond. $4303.47$
Root an. cond. $8.09943$
Motivic weight $7$
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  − 16·2-s − 54·3-s + 192·4-s + 250·5-s + 864·6-s + 686·7-s − 2.04e3·8-s + 2.18e3·9-s − 4.00e3·10-s + 4.72e3·11-s − 1.03e4·12-s + 1.06e4·13-s − 1.09e4·14-s − 1.35e4·15-s + 2.04e4·16-s + 2.78e4·17-s − 3.49e4·18-s + 2.54e4·19-s + 4.80e4·20-s − 3.70e4·21-s − 7.56e4·22-s + 8.02e4·23-s + 1.10e5·24-s + 4.68e4·25-s − 1.69e5·26-s − 7.87e4·27-s + 1.31e5·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.07·11-s − 1.73·12-s + 1.33·13-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 1.37·17-s − 1.41·18-s + 0.850·19-s + 1.34·20-s − 0.872·21-s − 1.51·22-s + 1.37·23-s + 1.63·24-s + 3/5·25-s − 1.89·26-s − 0.769·27-s + 1.13·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(44100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(4303.47\)
Root analytic conductor: \(8.09943\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 44100,\ (\ :7/2, 7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.512329251\)
\(L(\frac12)\) \(\approx\) \(2.512329251\)
\(L(\frac{9}{2})\) 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$C_1$ \( ( 1 + p^{3} T )^{2} \)
3$C_1$ \( ( 1 + p^{3} T )^{2} \)
5$C_1$ \( ( 1 - p^{3} T )^{2} \)
7$C_1$ \( ( 1 - p^{3} T )^{2} \)
good11$D_{4}$ \( 1 - 4728 T + 34054294 T^{2} - 4728 p^{7} T^{3} + p^{14} T^{4} \)
13$D_{4}$ \( 1 - 10612 T + 74179806 T^{2} - 10612 p^{7} T^{3} + p^{14} T^{4} \)
17$D_{4}$ \( 1 - 27852 T + 462255478 T^{2} - 27852 p^{7} T^{3} + p^{14} T^{4} \)
19$D_{4}$ \( 1 - 25432 T + 1358334534 T^{2} - 25432 p^{7} T^{3} + p^{14} T^{4} \)
23$D_{4}$ \( 1 - 80232 T + 5644031950 T^{2} - 80232 p^{7} T^{3} + p^{14} T^{4} \)
29$D_{4}$ \( 1 + 45324 T - 7026768194 T^{2} + 45324 p^{7} T^{3} + p^{14} T^{4} \)
31$D_{4}$ \( 1 + 7352 T + 55032830142 T^{2} + 7352 p^{7} T^{3} + p^{14} T^{4} \)
37$D_{4}$ \( 1 + 205820 T + 200374751502 T^{2} + 205820 p^{7} T^{3} + p^{14} T^{4} \)
41$D_{4}$ \( 1 + 36684 T - 7695869930 T^{2} + 36684 p^{7} T^{3} + p^{14} T^{4} \)
43$D_{4}$ \( 1 + 1080320 T + 763452135990 T^{2} + 1080320 p^{7} T^{3} + p^{14} T^{4} \)
47$D_{4}$ \( 1 + 718344 T + 720857495326 T^{2} + 718344 p^{7} T^{3} + p^{14} T^{4} \)
53$D_{4}$ \( 1 + 1913172 T + 3257645874334 T^{2} + 1913172 p^{7} T^{3} + p^{14} T^{4} \)
59$D_{4}$ \( 1 - 1895976 T + 5812867275382 T^{2} - 1895976 p^{7} T^{3} + p^{14} T^{4} \)
61$D_{4}$ \( 1 + 963356 T + 4022639008542 T^{2} + 963356 p^{7} T^{3} + p^{14} T^{4} \)
67$D_{4}$ \( 1 - 645712 T + 8915451427782 T^{2} - 645712 p^{7} T^{3} + p^{14} T^{4} \)
71$D_{4}$ \( 1 - 9304296 T + 39832610334190 T^{2} - 9304296 p^{7} T^{3} + p^{14} T^{4} \)
73$D_{4}$ \( 1 + 41492 T + 21281623775814 T^{2} + 41492 p^{7} T^{3} + p^{14} T^{4} \)
79$D_{4}$ \( 1 - 2904736 T + 36253149389406 T^{2} - 2904736 p^{7} T^{3} + p^{14} T^{4} \)
83$D_{4}$ \( 1 + 1393800 T + 23867798788678 T^{2} + 1393800 p^{7} T^{3} + p^{14} T^{4} \)
89$D_{4}$ \( 1 - 5532180 T + 95885297168758 T^{2} - 5532180 p^{7} T^{3} + p^{14} T^{4} \)
97$D_{4}$ \( 1 - 8275564 T + 177218291492886 T^{2} - 8275564 p^{7} T^{3} + p^{14} T^{4} \)
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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

−11.10769087403262710186482644178, −11.02124252890975536411986895794, −10.19631463553942171038005838015, −9.984927320354226730473176505945, −9.419552470061934363110871258191, −9.003683309163803619388401213623, −8.412938889605492049487065120024, −7.949192467310936185921087630355, −7.26562782712374972902170745237, −6.75668566602880491960906779263, −6.31716521496178446730058043013, −5.87216948477884351398340986446, −5.13419070896361650328405435871, −4.90458982056405765625894297727, −3.47048793894981845308349680303, −3.35557189516235031806206583239, −1.82460156635262319945314002737, −1.67446026356516064973994833031, −0.894155528607125989807010925029, −0.73713980284009799641292783297, 0.73713980284009799641292783297, 0.894155528607125989807010925029, 1.67446026356516064973994833031, 1.82460156635262319945314002737, 3.35557189516235031806206583239, 3.47048793894981845308349680303, 4.90458982056405765625894297727, 5.13419070896361650328405435871, 5.87216948477884351398340986446, 6.31716521496178446730058043013, 6.75668566602880491960906779263, 7.26562782712374972902170745237, 7.949192467310936185921087630355, 8.412938889605492049487065120024, 9.003683309163803619388401213623, 9.419552470061934363110871258191, 9.984927320354226730473176505945, 10.19631463553942171038005838015, 11.02124252890975536411986895794, 11.10769087403262710186482644178

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