Properties

Label 4-882e2-1.1-c2e2-0-2
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $577.573$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 30·13-s + 4·16-s + 26·19-s + 48·25-s − 6·31-s + 34·37-s − 170·43-s + 60·52-s + 144·61-s − 8·64-s + 86·67-s + 190·73-s − 52·76-s + 138·79-s − 32·97-s − 96·100-s + 122·103-s − 130·109-s + 192·121-s + 12·124-s + 127-s + 131-s + 137-s + 139-s − 68·148-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s − 2.30·13-s + 1/4·16-s + 1.36·19-s + 1.91·25-s − 0.193·31-s + 0.918·37-s − 3.95·43-s + 1.15·52-s + 2.36·61-s − 1/8·64-s + 1.28·67-s + 2.60·73-s − 0.684·76-s + 1.74·79-s − 0.329·97-s − 0.959·100-s + 1.18·103-s − 1.19·109-s + 1.58·121-s + 3/31·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 0.459·148-s + 0.00671·149-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(577.573\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.551015968\)
\(L(\frac12)\) \(\approx\) \(1.551015968\)
\(L(2)\) 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$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 48 T^{2} + p^{4} T^{4} \)
11$C_2^2$ \( 1 - 192 T^{2} + p^{4} T^{4} \)
13$C_2$ \( ( 1 + 15 T + p^{2} T^{2} )^{2} \)
17$C_2^2$ \( 1 - 450 T^{2} + p^{4} T^{4} \)
19$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 546 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 1170 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 + 3 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 - 17 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 + 3136 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 85 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 + 784 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4466 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 + 1230 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 - 72 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 - 43 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 - 7344 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 95 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 69 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 10080 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 + 2590 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
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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

−9.941460077108236694753244515272, −9.743405561536101881666670195893, −9.501854997083838322202655219409, −8.958656233103028161858077082397, −8.371575074348962539815112467748, −8.117301664371589118834300451148, −7.63800286154766170964470094604, −7.12313120230937540786521166770, −6.70763316536792190446646403702, −6.55391027624631925372186795301, −5.50660914541027995979175712572, −5.11559708713634202278128079984, −4.99266264363685669539654090258, −4.58266151970502913797721773724, −3.57271242295702705018898993639, −3.41639324905342259393104321056, −2.57629519634764485108023629744, −2.20227081949867521130733780516, −1.17401070769927964154356917892, −0.45613760243433685964316918352, 0.45613760243433685964316918352, 1.17401070769927964154356917892, 2.20227081949867521130733780516, 2.57629519634764485108023629744, 3.41639324905342259393104321056, 3.57271242295702705018898993639, 4.58266151970502913797721773724, 4.99266264363685669539654090258, 5.11559708713634202278128079984, 5.50660914541027995979175712572, 6.55391027624631925372186795301, 6.70763316536792190446646403702, 7.12313120230937540786521166770, 7.63800286154766170964470094604, 8.117301664371589118834300451148, 8.371575074348962539815112467748, 8.958656233103028161858077082397, 9.501854997083838322202655219409, 9.743405561536101881666670195893, 9.941460077108236694753244515272

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