Properties

Label 4-24e4-1.1-c6e2-0-2
Degree $4$
Conductor $331776$
Sign $1$
Analytic cond. $17559.2$
Root an. cond. $11.5113$
Motivic weight $6$
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  − 968·7-s − 6.73e3·13-s − 1.14e4·19-s + 992·25-s − 7.95e4·31-s − 1.05e5·37-s − 7.60e3·43-s + 4.67e5·49-s − 2.65e4·61-s − 3.37e5·67-s + 4.72e5·73-s − 7.02e4·79-s + 6.52e6·91-s − 6.42e5·97-s + 3.98e6·103-s − 3.88e5·109-s + 1.74e6·121-s + 127-s + 131-s + 1.11e7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 2.82·7-s − 3.06·13-s − 1.67·19-s + 0.0634·25-s − 2.67·31-s − 2.07·37-s − 0.0955·43-s + 3.97·49-s − 0.116·61-s − 1.12·67-s + 1.21·73-s − 0.142·79-s + 8.65·91-s − 0.704·97-s + 3.64·103-s − 0.300·109-s + 0.985·121-s + 4.72·133-s + 5.05·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(331776\)    =    \(2^{12} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(17559.2\)
Root analytic conductor: \(11.5113\)
Motivic weight: \(6\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{576} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 331776,\ (\ :3, 3),\ 1)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.01743197464\)
\(L(\frac12)\) \(\approx\) \(0.01743197464\)
\(L(4)\) 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 \( 1 \)
good5$C_2^2$ \( 1 - 992 T^{2} + p^{12} T^{4} \)
7$C_2$ \( ( 1 + 484 T + p^{6} T^{2} )^{2} \)
11$C_2^2$ \( 1 - 1745714 T^{2} + p^{12} T^{4} \)
13$C_2$ \( ( 1 + 3368 T + p^{6} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 48274976 T^{2} + p^{12} T^{4} \)
19$C_2$ \( ( 1 + 5744 T + p^{6} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 284666690 T^{2} + p^{12} T^{4} \)
29$C_2^2$ \( 1 - 327940544 T^{2} + p^{12} T^{4} \)
31$C_2$ \( ( 1 + 39796 T + p^{6} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 52526 T + p^{6} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 8128061984 T^{2} + p^{12} T^{4} \)
43$C_2$ \( ( 1 + 3800 T + p^{6} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 15661450658 T^{2} + p^{12} T^{4} \)
53$C_2^2$ \( 1 + 12666935680 T^{2} + p^{12} T^{4} \)
59$C_2^2$ \( 1 - 21940729490 T^{2} + p^{12} T^{4} \)
61$C_2$ \( ( 1 + 13250 T + p^{6} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 168968 T + p^{6} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 26256724990 T^{2} + p^{12} T^{4} \)
73$C_2$ \( ( 1 - 236144 T + p^{6} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 35116 T + p^{6} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 653760187346 T^{2} + p^{12} T^{4} \)
89$C_2^2$ \( 1 - 977236742720 T^{2} + p^{12} T^{4} \)
97$C_2$ \( ( 1 + 321424 T + p^{6} T^{2} )^{2} \)
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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

−10.27876645180557098915012920921, −9.877492568020404259030051185343, −9.094250275509000795959074762102, −8.985396480364529280009836643966, −8.321870104776695269627931387523, −7.37436806806884325193501357667, −7.22028686230430019203986614233, −7.05743335762494203404048075634, −6.27210631400804706917432337395, −6.17520176130796224768196222728, −5.33529193464473251429447855219, −5.03548391886707498778162653087, −4.33939507987719952692668634255, −3.72432729673021651103889937211, −3.26504092334992315888035816923, −2.87513367723826408985859619115, −2.05765000041784171102140571965, −2.02985928224993188831161217903, −0.49579283441015451900999534308, −0.04823688285102974160036775537, 0.04823688285102974160036775537, 0.49579283441015451900999534308, 2.02985928224993188831161217903, 2.05765000041784171102140571965, 2.87513367723826408985859619115, 3.26504092334992315888035816923, 3.72432729673021651103889937211, 4.33939507987719952692668634255, 5.03548391886707498778162653087, 5.33529193464473251429447855219, 6.17520176130796224768196222728, 6.27210631400804706917432337395, 7.05743335762494203404048075634, 7.22028686230430019203986614233, 7.37436806806884325193501357667, 8.321870104776695269627931387523, 8.985396480364529280009836643966, 9.094250275509000795959074762102, 9.877492568020404259030051185343, 10.27876645180557098915012920921

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