Properties

Label 4-882e2-1.1-c3e2-0-6
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

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 12·4-s + 5·5-s − 32·8-s − 20·10-s − 67·11-s − 41·13-s + 80·16-s − 92·17-s − 43·19-s + 60·20-s + 268·22-s − 148·23-s + 105·25-s + 164·26-s − 77·29-s + 520·31-s − 192·32-s + 368·34-s + 7·37-s + 172·38-s − 160·40-s − 426·41-s − 107·43-s − 804·44-s + 592·46-s + 576·47-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s + 0.447·5-s − 1.41·8-s − 0.632·10-s − 1.83·11-s − 0.874·13-s + 5/4·16-s − 1.31·17-s − 0.519·19-s + 0.670·20-s + 2.59·22-s − 1.34·23-s + 0.839·25-s + 1.23·26-s − 0.493·29-s + 3.01·31-s − 1.06·32-s + 1.85·34-s + 0.0311·37-s + 0.734·38-s − 0.632·40-s − 1.62·41-s − 0.379·43-s − 2.75·44-s + 1.89·46-s + 1.78·47-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: \(0\)
Selberg data: \((4,\ 777924,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5022438307\)
\(L(\frac12)\) \(\approx\) \(0.5022438307\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 - p T - 16 p T^{2} - p^{4} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 + 67 T + 3448 T^{2} + 67 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 + 92 T + 386 p T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 + 43 T + 11154 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 + 148 T + 24430 T^{2} + 148 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 + 77 T + 9574 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 - 520 T + 125837 T^{2} - 520 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 7 T + 74082 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 576 T + 242170 T^{2} - 576 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 243 T + 309490 T^{2} - 243 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 + 7 T + 200614 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 224 T + 461126 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 687 T + 678832 T^{2} - 687 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 - 921 T + 578188 T^{2} - 921 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 + 526 T + 1033727 T^{2} + 526 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 774 T + 966562 T^{2} - 774 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} \)
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

−9.920760678026263954522944325170, −9.778954434501738103413008337758, −9.001764440803159914618674708363, −8.687005971344750862646124584750, −8.361176961739419916938766065822, −7.893452890775865171720399343145, −7.65111755265060495970696936680, −7.03099133091059847917472018679, −6.55687947172848721711430682831, −6.39035258100666664206190810275, −5.58378454694971852449732218219, −5.32852797218403187572843427200, −4.57788457963912621757636169833, −4.30722919275791766187054369466, −3.21461499887703329325054823246, −2.73802504144092160208485812766, −2.12676951194245271550189996032, −2.09407121541824752779551192959, −0.897999339228385798841083041356, −0.27849165999967960918650292112, 0.27849165999967960918650292112, 0.897999339228385798841083041356, 2.09407121541824752779551192959, 2.12676951194245271550189996032, 2.73802504144092160208485812766, 3.21461499887703329325054823246, 4.30722919275791766187054369466, 4.57788457963912621757636169833, 5.32852797218403187572843427200, 5.58378454694971852449732218219, 6.39035258100666664206190810275, 6.55687947172848721711430682831, 7.03099133091059847917472018679, 7.65111755265060495970696936680, 7.893452890775865171720399343145, 8.361176961739419916938766065822, 8.687005971344750862646124584750, 9.001764440803159914618674708363, 9.778954434501738103413008337758, 9.920760678026263954522944325170

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