Properties

Label 4-21e4-1.1-c1e2-0-1
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $12.4002$
Root an. cond. $1.87654$
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  + 2·2-s − 4-s − 5-s − 8·8-s − 3·9-s − 2·10-s − 5·11-s − 5·13-s − 7·16-s + 3·17-s − 6·18-s + 19-s + 20-s − 10·22-s − 3·23-s + 5·25-s − 10·26-s + 29-s + 14·32-s + 6·34-s + 3·36-s − 3·37-s + 2·38-s + 8·40-s − 5·41-s + 43-s + 5·44-s + ⋯
L(s)  = 1  + 1.41·2-s − 1/2·4-s − 0.447·5-s − 2.82·8-s − 9-s − 0.632·10-s − 1.50·11-s − 1.38·13-s − 7/4·16-s + 0.727·17-s − 1.41·18-s + 0.229·19-s + 0.223·20-s − 2.13·22-s − 0.625·23-s + 25-s − 1.96·26-s + 0.185·29-s + 2.47·32-s + 1.02·34-s + 1/2·36-s − 0.493·37-s + 0.324·38-s + 1.26·40-s − 0.780·41-s + 0.152·43-s + 0.753·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(12.4002\)
Root analytic conductor: \(1.87654\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8103065141\)
\(L(\frac12)\) \(\approx\) \(0.8103065141\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + p T^{2} \)
7 \( 1 \)
good2$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2^2$ \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
41$C_2^2$ \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 13 T + 80 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
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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

−11.77571165427523761745811034438, −11.01144369866925413890849561797, −10.40692253208948842615237675603, −9.891365905234033957090596478698, −9.759990293972023114358226004503, −8.947118938118963554793137628289, −8.647589038592275372985870014402, −8.050452790144962242927775413890, −7.942170815032265332943663847739, −7.01334971871965931888962796787, −6.63708244465516104498097115985, −5.64282061840593164526493357085, −5.34611626000252588588114924133, −5.33256860122687064871953288635, −4.62499470423962819147532538854, −4.04673028940956616442490297138, −3.49025161554665098994571218138, −2.76867154788269676611066478745, −2.58159974912724903515731153491, −0.44116309322243106104866047721, 0.44116309322243106104866047721, 2.58159974912724903515731153491, 2.76867154788269676611066478745, 3.49025161554665098994571218138, 4.04673028940956616442490297138, 4.62499470423962819147532538854, 5.33256860122687064871953288635, 5.34611626000252588588114924133, 5.64282061840593164526493357085, 6.63708244465516104498097115985, 7.01334971871965931888962796787, 7.942170815032265332943663847739, 8.050452790144962242927775413890, 8.647589038592275372985870014402, 8.947118938118963554793137628289, 9.759990293972023114358226004503, 9.891365905234033957090596478698, 10.40692253208948842615237675603, 11.01144369866925413890849561797, 11.77571165427523761745811034438

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