Properties

Label 4-684e2-1.1-c4e2-0-2
Degree $4$
Conductor $467856$
Sign $1$
Analytic cond. $4999.20$
Root an. cond. $8.40862$
Motivic weight $4$
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  + 31·5-s + 73·7-s − 233·11-s − 353·17-s + 722·19-s + 316·23-s + 625·25-s + 2.26e3·35-s − 3.52e3·43-s + 1.20e3·47-s + 2.40e3·49-s − 7.22e3·55-s − 3.16e3·61-s + 1.00e4·73-s − 1.70e4·77-s + 1.13e4·83-s − 1.09e4·85-s + 2.23e4·95-s + 1.99e4·101-s + 9.79e3·115-s − 2.57e4·119-s + 1.46e4·121-s + 2.83e4·125-s + 127-s + 131-s + 5.27e4·133-s + 137-s + ⋯
L(s)  = 1  + 1.23·5-s + 1.48·7-s − 1.92·11-s − 1.22·17-s + 2·19-s + 0.597·23-s + 25-s + 1.84·35-s − 1.90·43-s + 0.546·47-s + 49-s − 2.38·55-s − 0.851·61-s + 1.88·73-s − 2.86·77-s + 1.64·83-s − 1.51·85-s + 2.47·95-s + 1.96·101-s + 0.740·115-s − 1.81·119-s + 121-s + 1.81·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 2.97·133-s + 5.32e−5·137-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(467856\)    =    \(2^{4} \cdot 3^{4} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(4999.20\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{684} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 467856,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(4.364783957\)
\(L(\frac12)\) \(\approx\) \(4.364783957\)
\(L(3)\) 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 \( 1 \)
3 \( 1 \)
19$C_1$ \( ( 1 - p^{2} T )^{2} \)
good5$C_2^2$ \( 1 - 31 T + 336 T^{2} - 31 p^{4} T^{3} + p^{8} T^{4} \)
7$C_2^2$ \( 1 - 73 T + 2928 T^{2} - 73 p^{4} T^{3} + p^{8} T^{4} \)
11$C_2^2$ \( 1 + 233 T + 39648 T^{2} + 233 p^{4} T^{3} + p^{8} T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
17$C_2^2$ \( 1 + 353 T + 41088 T^{2} + 353 p^{4} T^{3} + p^{8} T^{4} \)
23$C_2$ \( ( 1 - 158 T + p^{4} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
31$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
43$C_2^2$ \( 1 + 3527 T + 9020928 T^{2} + 3527 p^{4} T^{3} + p^{8} T^{4} \)
47$C_2^2$ \( 1 - 1207 T - 3422832 T^{2} - 1207 p^{4} T^{3} + p^{8} T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
61$C_2^2$ \( 1 + 3167 T - 3815952 T^{2} + 3167 p^{4} T^{3} + p^{8} T^{4} \)
67$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
73$C_2^2$ \( 1 - 10033 T + 72262848 T^{2} - 10033 p^{4} T^{3} + p^{8} T^{4} \)
79$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
83$C_2$ \( ( 1 - 5678 T + p^{4} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \)
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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

−10.02253749190451278162832118078, −9.863536444905527087454691198256, −9.097713414974647512888263573643, −9.001606123288995156181785979244, −8.158188547839796509376966679581, −8.140664795171600896140142054606, −7.41380428436601002106888725730, −7.25563937610249660795790106054, −6.48682899923729606214334770842, −6.07374671003484679137551808463, −5.29018141071106057729556462726, −5.14410936544971524311460648739, −4.99733779350976551708115986404, −4.31251386230859247418610016906, −3.32653012406943954002177611638, −2.91715570505957179991071444979, −2.16969078158148094638051052078, −1.94529827620458434947262888100, −1.18351352672734375070448118228, −0.50850535342396402023769706967, 0.50850535342396402023769706967, 1.18351352672734375070448118228, 1.94529827620458434947262888100, 2.16969078158148094638051052078, 2.91715570505957179991071444979, 3.32653012406943954002177611638, 4.31251386230859247418610016906, 4.99733779350976551708115986404, 5.14410936544971524311460648739, 5.29018141071106057729556462726, 6.07374671003484679137551808463, 6.48682899923729606214334770842, 7.25563937610249660795790106054, 7.41380428436601002106888725730, 8.140664795171600896140142054606, 8.158188547839796509376966679581, 9.001606123288995156181785979244, 9.097713414974647512888263573643, 9.863536444905527087454691198256, 10.02253749190451278162832118078

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