Properties

Label 4-21e4-1.1-c3e2-0-10
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $677.032$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 8·4-s − 4·5-s + 32·8-s − 16·10-s + 62·11-s − 124·13-s + 128·16-s + 84·17-s − 100·19-s − 32·20-s + 248·22-s − 42·23-s + 125·25-s − 496·26-s + 20·29-s + 48·31-s + 256·32-s + 336·34-s + 246·37-s − 400·38-s − 128·40-s + 496·41-s + 136·43-s + 496·44-s − 168·46-s + 324·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s − 0.357·5-s + 1.41·8-s − 0.505·10-s + 1.69·11-s − 2.64·13-s + 2·16-s + 1.19·17-s − 1.20·19-s − 0.357·20-s + 2.40·22-s − 0.380·23-s + 25-s − 3.74·26-s + 0.128·29-s + 0.278·31-s + 1.41·32-s + 1.69·34-s + 1.09·37-s − 1.70·38-s − 0.505·40-s + 1.88·41-s + 0.482·43-s + 1.69·44-s − 0.538·46-s + 1.00·47-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: $1$
Analytic conductor: \(677.032\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
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

\(L(2)\) \(\approx\) \(6.544978311\)
\(L(\frac12)\) \(\approx\) \(6.544978311\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 - 62 T + 2513 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 62 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 100 T + 3141 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 42 T - 10403 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 10 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 48 T - 27487 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 248 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 68 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 324 T + 1153 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 258 T - 82313 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 120 T - 190979 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 622 T + 159903 T^{2} + 622 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 904 T + 516453 T^{2} + 904 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 678 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 642 T + 23147 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 740 T + 54561 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 468 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 200 T - 664969 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 1266 T + p^{3} T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.84327885673245217543925897569, −10.73009130820586390128416074635, −10.07815060372108204938249646774, −9.587771310636884164334517806318, −9.337394503780624791968877700012, −8.632785735193207637240522322908, −7.78160001198494247423809153835, −7.77047382940256863652728793137, −7.04261703421583993768362908340, −6.83468802252789932172451463895, −6.05851747561600477167807362356, −5.62841246532407671667947239360, −5.01609353733475628478868443301, −4.43454341049976369631177694198, −4.26674923330580716486259309958, −3.78079238597213965929770116505, −2.83372343395592904218394437471, −2.44434089531320147633408604591, −1.49572059144149677364253525177, −0.68434121406074016123907004078, 0.68434121406074016123907004078, 1.49572059144149677364253525177, 2.44434089531320147633408604591, 2.83372343395592904218394437471, 3.78079238597213965929770116505, 4.26674923330580716486259309958, 4.43454341049976369631177694198, 5.01609353733475628478868443301, 5.62841246532407671667947239360, 6.05851747561600477167807362356, 6.83468802252789932172451463895, 7.04261703421583993768362908340, 7.77047382940256863652728793137, 7.78160001198494247423809153835, 8.632785735193207637240522322908, 9.337394503780624791968877700012, 9.587771310636884164334517806318, 10.07815060372108204938249646774, 10.73009130820586390128416074635, 10.84327885673245217543925897569

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