Properties

Label 4-882e2-1.1-c3e2-0-2
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $2708.12$
Root an. cond. $7.21385$
Motivic weight $3$
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  + 2·2-s + 18·5-s − 8·8-s + 36·10-s − 72·11-s − 68·13-s − 16·16-s + 6·17-s − 92·19-s − 144·22-s − 180·23-s + 125·25-s − 136·26-s + 228·29-s − 56·31-s + 12·34-s + 34·37-s − 184·38-s − 144·40-s − 12·41-s + 328·43-s − 360·46-s + 168·47-s + 250·50-s + 654·53-s − 1.29e3·55-s + 456·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.60·5-s − 0.353·8-s + 1.13·10-s − 1.97·11-s − 1.45·13-s − 1/4·16-s + 0.0856·17-s − 1.11·19-s − 1.39·22-s − 1.63·23-s + 25-s − 1.02·26-s + 1.45·29-s − 0.324·31-s + 0.0605·34-s + 0.151·37-s − 0.785·38-s − 0.569·40-s − 0.0457·41-s + 1.16·43-s − 1.15·46-s + 0.521·47-s + 0.707·50-s + 1.69·53-s − 3.17·55-s + 1.03·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(2708.12\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9174184789\)
\(L(\frac12)\) \(\approx\) \(0.9174184789\)
\(L(\frac{5}{2})\) 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)$
bad2$C_2$ \( 1 - p T + p^{2} T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 18 T + 199 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 72 T + 3853 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 + 34 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 6 T - 4877 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 92 T + 1605 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 + 180 T + 20233 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 114 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 56 T - 26655 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 - 34 T - 49497 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 164 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 168 T - 75599 T^{2} - 168 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 654 T + 278839 T^{2} - 654 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 + 492 T + 36685 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 250 T - 164481 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 - 124 T - 285387 T^{2} - 124 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 36 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 1010 T + 631083 T^{2} + 1010 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 + 56 T - 489903 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 + 228 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 390 T - 552869 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 70 T + p^{3} T^{2} )^{2} \)
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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

−10.17682544642988609784279784092, −9.752242672255233194586873788912, −9.257968920093157126934108415690, −8.505714398240621808622127694839, −8.486706441210684143969890904634, −7.76184574712418731343228700030, −7.36705970885382882934481207078, −6.97043644373100683648991830127, −6.17035580961689196971398604650, −5.97738217449239003240660322160, −5.64520922713815368824224805549, −5.09195024477580779313941494511, −4.76860338228248145037761451940, −4.29705270524887573387812515617, −3.64810735599808363173423859172, −2.66849887594868981598694219678, −2.42791625958768324447019374584, −2.30402539244221508885544116223, −1.33184794008812499500985889060, −0.20559362910245983440748026613, 0.20559362910245983440748026613, 1.33184794008812499500985889060, 2.30402539244221508885544116223, 2.42791625958768324447019374584, 2.66849887594868981598694219678, 3.64810735599808363173423859172, 4.29705270524887573387812515617, 4.76860338228248145037761451940, 5.09195024477580779313941494511, 5.64520922713815368824224805549, 5.97738217449239003240660322160, 6.17035580961689196971398604650, 6.97043644373100683648991830127, 7.36705970885382882934481207078, 7.76184574712418731343228700030, 8.486706441210684143969890904634, 8.505714398240621808622127694839, 9.257968920093157126934108415690, 9.752242672255233194586873788912, 10.17682544642988609784279784092

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