Properties

Label 4-1840e2-1.1-c1e2-0-3
Degree $4$
Conductor $3385600$
Sign $1$
Analytic cond. $215.868$
Root an. cond. $3.83307$
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  − 3·3-s + 2·5-s − 3·7-s + 4·9-s + 7·11-s + 3·13-s − 6·15-s + 3·17-s − 19-s + 9·21-s + 2·23-s + 3·25-s − 6·27-s + 2·29-s + 5·31-s − 21·33-s − 6·35-s + 16·37-s − 9·39-s − 9·41-s + 4·43-s + 8·45-s − 2·47-s − 4·49-s − 9·51-s − 8·53-s + 14·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.894·5-s − 1.13·7-s + 4/3·9-s + 2.11·11-s + 0.832·13-s − 1.54·15-s + 0.727·17-s − 0.229·19-s + 1.96·21-s + 0.417·23-s + 3/5·25-s − 1.15·27-s + 0.371·29-s + 0.898·31-s − 3.65·33-s − 1.01·35-s + 2.63·37-s − 1.44·39-s − 1.40·41-s + 0.609·43-s + 1.19·45-s − 0.291·47-s − 4/7·49-s − 1.26·51-s − 1.09·53-s + 1.88·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3385600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3385600\)    =    \(2^{8} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(215.868\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1840} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3385600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892675641\)
\(L(\frac12)\) \(\approx\) \(1.892675641\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$C_4$ \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 7 T + 31 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + T + 9 T^{2} + p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 5 T + 39 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 5 T + 47 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 - 29 T + 349 T^{2} - 29 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 9 T + 211 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
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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

−9.639518072108109095119073302977, −9.356191689125399719788492406650, −8.559769195233725667887525447222, −8.511630220375591229806182939849, −7.83763763444015083861030807599, −7.25137285362637379695327845881, −6.60651191031257657139381908263, −6.54854135794469658397589842182, −6.18822667182504150839786403064, −6.18235394571940089081014721812, −5.45130256126043818109353580816, −5.18966356712349264863638440698, −4.64017857004180320368311422733, −4.07045182124824515884879004844, −3.62099934720517574066281107058, −3.28323056710320615391656462273, −2.44607867909984537662334278063, −1.78870221096584735277916308684, −0.888337481180660831928006609144, −0.836472660722266300355698285920, 0.836472660722266300355698285920, 0.888337481180660831928006609144, 1.78870221096584735277916308684, 2.44607867909984537662334278063, 3.28323056710320615391656462273, 3.62099934720517574066281107058, 4.07045182124824515884879004844, 4.64017857004180320368311422733, 5.18966356712349264863638440698, 5.45130256126043818109353580816, 6.18235394571940089081014721812, 6.18822667182504150839786403064, 6.54854135794469658397589842182, 6.60651191031257657139381908263, 7.25137285362637379695327845881, 7.83763763444015083861030807599, 8.511630220375591229806182939849, 8.559769195233725667887525447222, 9.356191689125399719788492406650, 9.639518072108109095119073302977

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