Properties

Degree $4$
Conductor $194481$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 8·4-s + 18·5-s − 45·8-s − 54·10-s − 36·11-s + 68·13-s + 135·16-s − 42·17-s − 124·19-s + 144·20-s + 108·22-s + 125·25-s − 204·26-s − 204·29-s − 160·31-s − 360·32-s + 126·34-s − 398·37-s + 372·38-s − 810·40-s − 636·41-s − 536·43-s − 288·44-s − 240·47-s − 375·50-s + 544·52-s + ⋯
L(s)  = 1  − 1.06·2-s + 4-s + 1.60·5-s − 1.98·8-s − 1.70·10-s − 0.986·11-s + 1.45·13-s + 2.10·16-s − 0.599·17-s − 1.49·19-s + 1.60·20-s + 1.04·22-s + 25-s − 1.53·26-s − 1.30·29-s − 0.926·31-s − 1.98·32-s + 0.635·34-s − 1.76·37-s + 1.58·38-s − 3.20·40-s − 2.42·41-s − 1.90·43-s − 0.986·44-s − 0.744·47-s − 1.06·50-s + 1.45·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(0.3149783717\)
\(L(\frac12)\) \(\approx\) \(0.3149783717\)
\(L(\frac{5}{2})\) 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)$
bad3 \( 1 \)
7 \( 1 \)
good2$C_2^2$ \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 36 T - 35 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 34 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 42 T - 3149 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 124 T + 8517 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - p^{3} T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 102 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 160 T - 4191 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 398 T + 107751 T^{2} + 398 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 318 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 + 268 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 240 T - 46223 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 + 498 T + 99127 T^{2} + 498 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 132 T - 187955 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 398 T - 68577 T^{2} - 398 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 92 T - 292299 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 720 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 502 T - 137013 T^{2} + 502 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 1024 T + 555537 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 204 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 + 354 T - 579653 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 - 286 T + p^{3} T^{2} )^{2} \)
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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

−10.73783619237473958207888739322, −10.35750882434308464184856416558, −10.08460477547032023170846694478, −9.456545376130456516394900418410, −9.245959524958729161291051045576, −8.593546229540331771165171535595, −8.411322810560292227992972926212, −8.079313455730280518434742586027, −7.06954791184809672158301334054, −6.58114522080549294565524718101, −6.46526332292883172555140169716, −5.83515999586349296442664760332, −5.35915765392515151868403210278, −5.02618390217363575956534735582, −3.61128950994451395853673891524, −3.45759249559648530847212653071, −2.50271471630768336240804782313, −1.77620100138586813884053069954, −1.72437727461189320406063306224, −0.19888443130704633358617067793, 0.19888443130704633358617067793, 1.72437727461189320406063306224, 1.77620100138586813884053069954, 2.50271471630768336240804782313, 3.45759249559648530847212653071, 3.61128950994451395853673891524, 5.02618390217363575956534735582, 5.35915765392515151868403210278, 5.83515999586349296442664760332, 6.46526332292883172555140169716, 6.58114522080549294565524718101, 7.06954791184809672158301334054, 8.079313455730280518434742586027, 8.411322810560292227992972926212, 8.593546229540331771165171535595, 9.245959524958729161291051045576, 9.456545376130456516394900418410, 10.08460477547032023170846694478, 10.35750882434308464184856416558, 10.73783619237473958207888739322

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