Properties

Label 4-882e2-1.1-c3e2-0-39
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 12·4-s + 32·8-s − 80·11-s + 80·16-s − 320·22-s − 136·23-s − 200·25-s − 220·29-s + 192·32-s − 40·37-s − 680·43-s − 960·44-s − 544·46-s − 800·50-s − 1.25e3·53-s − 880·58-s + 448·64-s + 1.08e3·67-s + 840·71-s − 160·74-s − 1.52e3·79-s − 2.72e3·86-s − 2.56e3·88-s − 1.63e3·92-s − 2.40e3·100-s − 5.02e3·106-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.41·8-s − 2.19·11-s + 5/4·16-s − 3.10·22-s − 1.23·23-s − 8/5·25-s − 1.40·29-s + 1.06·32-s − 0.177·37-s − 2.41·43-s − 3.28·44-s − 1.74·46-s − 2.26·50-s − 3.25·53-s − 1.99·58-s + 7/8·64-s + 1.96·67-s + 1.40·71-s − 0.251·74-s − 2.16·79-s − 3.41·86-s − 3.10·88-s − 1.84·92-s − 2.39·100-s − 4.60·106-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: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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$C_1$ \( ( 1 - p T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 8 p^{2} T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 40 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 344 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 9824 T^{2} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 13590 T^{2} + p^{6} T^{4} \)
23$C_2$ \( ( 1 + 68 T + p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 110 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 45470 T^{2} + p^{6} T^{4} \)
37$C_2$ \( ( 1 + 20 T + p^{3} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 135392 T^{2} + p^{6} T^{4} \)
43$C_2$ \( ( 1 + 340 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 199454 T^{2} + p^{6} T^{4} \)
53$C_2$ \( ( 1 + 628 T + p^{3} T^{2} )^{2} \)
59$C_2^2$ \( 1 - 358042 T^{2} + p^{6} T^{4} \)
61$C_2^2$ \( 1 - 388440 T^{2} + p^{6} T^{4} \)
67$C_2$ \( ( 1 - 540 T + p^{3} T^{2} )^{2} \)
71$C_2$ \( ( 1 - 420 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 693984 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 760 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 + 251126 T^{2} + p^{6} T^{4} \)
89$C_2^2$ \( 1 + 81488 T^{2} + p^{6} T^{4} \)
97$C_2^2$ \( 1 + 1573296 T^{2} + p^{6} T^{4} \)
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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

−9.725778701660704215345246628442, −9.344291069557412162470974285445, −8.315745238565526962526080568310, −8.134114786299807859530940129162, −7.79553157910202937552323822878, −7.50938534127597541854384262774, −6.66998073637732415002969678633, −6.55567738434376934896898272451, −5.80289680844946262702665755637, −5.51534287221672004091105183355, −5.12902023974388648388445275842, −4.79739450911468622155658371538, −3.90243426658539376455343675128, −3.85198287833411627621161539920, −3.01618959758702576407865535448, −2.67601088857169002793436145572, −1.90353166781167779560196388647, −1.66156170822587452417560838736, 0, 0, 1.66156170822587452417560838736, 1.90353166781167779560196388647, 2.67601088857169002793436145572, 3.01618959758702576407865535448, 3.85198287833411627621161539920, 3.90243426658539376455343675128, 4.79739450911468622155658371538, 5.12902023974388648388445275842, 5.51534287221672004091105183355, 5.80289680844946262702665755637, 6.55567738434376934896898272451, 6.66998073637732415002969678633, 7.50938534127597541854384262774, 7.79553157910202937552323822878, 8.134114786299807859530940129162, 8.315745238565526962526080568310, 9.344291069557412162470974285445, 9.725778701660704215345246628442

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