Properties

Label 4-21e4-1.1-c2e2-0-13
Degree $4$
Conductor $194481$
Sign $1$
Analytic cond. $144.393$
Root an. cond. $3.46646$
Motivic weight $2$
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  + 3·2-s + 6·3-s + 2·4-s + 18·6-s − 3·8-s + 27·9-s + 12·12-s + 4·13-s − 3·16-s − 27·17-s + 81·18-s − 11·19-s − 18·24-s + 38·25-s + 12·26-s + 108·27-s + 78·29-s − 32·31-s − 12·32-s − 81·34-s + 54·36-s + 34·37-s − 33·38-s + 24·39-s − 21·41-s + 61·43-s + 84·47-s + ⋯
L(s)  = 1  + 3/2·2-s + 2·3-s + 1/2·4-s + 3·6-s − 3/8·8-s + 3·9-s + 12-s + 4/13·13-s − 0.187·16-s − 1.58·17-s + 9/2·18-s − 0.578·19-s − 3/4·24-s + 1.51·25-s + 6/13·26-s + 4·27-s + 2.68·29-s − 1.03·31-s − 3/8·32-s − 2.38·34-s + 3/2·36-s + 0.918·37-s − 0.868·38-s + 8/13·39-s − 0.512·41-s + 1.41·43-s + 1.78·47-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(194481\)    =    \(3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(144.393\)
Root analytic conductor: \(3.46646\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 194481,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(10.94894767\)
\(L(\frac12)\) \(\approx\) \(10.94894767\)
\(L(2)\) 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$C_1$ \( ( 1 - p T )^{2} \)
7 \( 1 \)
good2$C_2^2$ \( 1 - 3 T + 7 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \)
5$C_2^2$ \( 1 - 38 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 239 T^{2} + p^{4} T^{4} \)
13$C_2^2$ \( 1 - 4 T - 153 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \)
17$C_2^2$ \( 1 + 27 T + 532 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 26 T + p^{2} T^{2} )( 1 + 37 T + p^{2} T^{2} ) \)
23$C_2^2$ \( 1 - 290 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 78 T + 2869 T^{2} - 78 p^{2} T^{3} + p^{4} T^{4} \)
31$C_2^2$ \( 1 + 32 T + 63 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} \)
37$C_2^2$ \( 1 - 34 T - 213 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \)
41$C_2^2$ \( 1 + 21 T + 1828 T^{2} + 21 p^{2} T^{3} + p^{4} T^{4} \)
43$C_2$ \( ( 1 - 83 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \)
47$C_2^2$ \( 1 - 84 T + 4561 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \)
53$C_2$ \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \)
59$C_2^2$ \( 1 + 87 T + 6004 T^{2} + 87 p^{2} T^{3} + p^{4} T^{4} \)
61$C_2^2$ \( 1 + 56 T - 585 T^{2} + 56 p^{2} T^{3} + p^{4} T^{4} \)
67$C_2^2$ \( 1 - 31 T - 3528 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \)
71$C_2^2$ \( 1 - 9110 T^{2} + p^{4} T^{4} \)
73$C_2^2$ \( 1 + 65 T - 1104 T^{2} + 65 p^{2} T^{3} + p^{4} T^{4} \)
79$C_2^2$ \( 1 + 38 T - 4797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} \)
83$C_2^2$ \( 1 + 84 T + 9241 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 216 T + 23473 T^{2} - 216 p^{2} T^{3} + p^{4} T^{4} \)
97$C_2^2$ \( 1 - 115 T + 3816 T^{2} - 115 p^{2} T^{3} + p^{4} T^{4} \)
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

−10.88412713466538810920506194542, −10.69000337665265712285985069629, −10.41390046773913804259601474445, −9.595204439592862508166623476297, −9.066159720185176860497601292033, −8.957398349723348297134739665787, −8.478287867491043460030038424428, −8.055842192656439783950212172647, −7.33927356908189366838490077343, −7.03816142807988026512299499909, −6.41026247929694589950661301747, −5.99312531683457480015699663656, −4.81162938268634913473921831495, −4.79802228635547893216331992136, −4.14584942411494333995207454940, −3.93911997160820473683707040781, −2.98572404861385715991670418901, −2.79794225313774067029137272078, −2.05347084757411291618423909921, −1.06559652123727098244201505184, 1.06559652123727098244201505184, 2.05347084757411291618423909921, 2.79794225313774067029137272078, 2.98572404861385715991670418901, 3.93911997160820473683707040781, 4.14584942411494333995207454940, 4.79802228635547893216331992136, 4.81162938268634913473921831495, 5.99312531683457480015699663656, 6.41026247929694589950661301747, 7.03816142807988026512299499909, 7.33927356908189366838490077343, 8.055842192656439783950212172647, 8.478287867491043460030038424428, 8.957398349723348297134739665787, 9.066159720185176860497601292033, 9.595204439592862508166623476297, 10.41390046773913804259601474445, 10.69000337665265712285985069629, 10.88412713466538810920506194542

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