Properties

Label 4-882e2-1.1-c3e2-0-31
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 12·4-s + 12·5-s + 32·8-s + 48·10-s + 4·11-s − 48·13-s + 80·16-s + 84·17-s + 72·19-s + 144·20-s + 16·22-s + 308·23-s − 44·25-s − 192·26-s + 80·29-s − 384·31-s + 192·32-s + 336·34-s + 536·37-s + 288·38-s + 384·40-s + 756·41-s + 400·43-s + 48·44-s + 1.23e3·46-s + 312·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s + 1.07·5-s + 1.41·8-s + 1.51·10-s + 0.109·11-s − 1.02·13-s + 5/4·16-s + 1.19·17-s + 0.869·19-s + 1.60·20-s + 0.155·22-s + 2.79·23-s − 0.351·25-s − 1.44·26-s + 0.512·29-s − 2.22·31-s + 1.06·32-s + 1.69·34-s + 2.38·37-s + 1.22·38-s + 1.51·40-s + 2.87·41-s + 1.41·43-s + 0.164·44-s + 3.94·46-s + 0.968·47-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: 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

\(L(2)\) \(\approx\) \(14.93813518\)
\(L(\frac12)\) \(\approx\) \(14.93813518\)
\(L(\frac{5}{2})\) 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$D_{4}$ \( 1 - 12 T + 188 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \)
11$D_{4}$ \( 1 - 4 T - 862 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \)
13$D_{4}$ \( 1 + 48 T + 4088 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \)
17$D_{4}$ \( 1 - 84 T + 676 p T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \)
19$D_{4}$ \( 1 - 72 T + 14622 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \)
23$D_{4}$ \( 1 - 308 T + 44522 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \)
29$D_{4}$ \( 1 - 80 T + 18626 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \)
31$D_{4}$ \( 1 + 384 T + 92918 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \)
37$D_{4}$ \( 1 - 536 T + 159018 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \)
41$D_{4}$ \( 1 - 756 T + 278276 T^{2} - 756 p^{3} T^{3} + p^{6} T^{4} \)
43$D_{4}$ \( 1 - 400 T + 142566 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \)
47$D_{4}$ \( 1 - 312 T + 222182 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \)
53$D_{4}$ \( 1 - 52 T + 241982 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \)
59$D_{4}$ \( 1 - 864 T + 596990 T^{2} - 864 p^{3} T^{3} + p^{6} T^{4} \)
61$D_{4}$ \( 1 + 1416 T + 938664 T^{2} + 1416 p^{3} T^{3} + p^{6} T^{4} \)
67$D_{4}$ \( 1 - 144 T + 550262 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \)
71$D_{4}$ \( 1 - 1524 T + 1208266 T^{2} - 1524 p^{3} T^{3} + p^{6} T^{4} \)
73$D_{4}$ \( 1 + 744 T + 241296 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \)
79$D_{4}$ \( 1 - 976 T + 532734 T^{2} - 976 p^{3} T^{3} + p^{6} T^{4} \)
83$D_{4}$ \( 1 + 312 T - 644698 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \)
89$D_{4}$ \( 1 - 108 T + 1330436 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} \)
97$D_{4}$ \( 1 - 744 T + 432480 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \)
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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

−9.864783963039341036748943054560, −9.556553922645206099061499193762, −9.178946293394297247613449360882, −9.073222236486308508142557890791, −7.899962458693330902949627752818, −7.69946421238976340555733443791, −7.23695780485060622098073355405, −7.06001676019769261432222178661, −6.19498659143027945997095611510, −5.96101408151034563092602807112, −5.43890016548073156820113994508, −5.32004243077272155792473801965, −4.65211902230538311190657271746, −4.24630351917497696373809426088, −3.55790985314990327580576680415, −3.05861659661790534892130539246, −2.42743037877594988650576148684, −2.29872978431818329262918064618, −1.09765747474199421275310300103, −0.944137019811539919812770333919, 0.944137019811539919812770333919, 1.09765747474199421275310300103, 2.29872978431818329262918064618, 2.42743037877594988650576148684, 3.05861659661790534892130539246, 3.55790985314990327580576680415, 4.24630351917497696373809426088, 4.65211902230538311190657271746, 5.32004243077272155792473801965, 5.43890016548073156820113994508, 5.96101408151034563092602807112, 6.19498659143027945997095611510, 7.06001676019769261432222178661, 7.23695780485060622098073355405, 7.69946421238976340555733443791, 7.899962458693330902949627752818, 9.073222236486308508142557890791, 9.178946293394297247613449360882, 9.556553922645206099061499193762, 9.864783963039341036748943054560

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