Properties

Label 4-33e2-1.1-c5e2-0-2
Degree $4$
Conductor $1089$
Sign $1$
Analytic cond. $28.0123$
Root an. cond. $2.30057$
Motivic weight $5$
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  + 13·2-s + 18·3-s + 71·4-s + 58·5-s + 234·6-s + 146·7-s + 65·8-s + 243·9-s + 754·10-s − 242·11-s + 1.27e3·12-s − 130·13-s + 1.89e3·14-s + 1.04e3·15-s − 1.72e3·16-s − 728·17-s + 3.15e3·18-s − 828·19-s + 4.11e3·20-s + 2.62e3·21-s − 3.14e3·22-s − 238·23-s + 1.17e3·24-s − 2.90e3·25-s − 1.69e3·26-s + 2.91e3·27-s + 1.03e4·28-s + ⋯
L(s)  = 1  + 2.29·2-s + 1.15·3-s + 2.21·4-s + 1.03·5-s + 2.65·6-s + 1.12·7-s + 0.359·8-s + 9-s + 2.38·10-s − 0.603·11-s + 2.56·12-s − 0.213·13-s + 2.58·14-s + 1.19·15-s − 1.68·16-s − 0.610·17-s + 2.29·18-s − 0.526·19-s + 2.30·20-s + 1.30·21-s − 1.38·22-s − 0.0938·23-s + 0.414·24-s − 0.928·25-s − 0.490·26-s + 0.769·27-s + 2.49·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(28.0123\)
Root analytic conductor: \(2.30057\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{33} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1089,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(10.13912971\)
\(L(\frac12)\) \(\approx\) \(10.13912971\)
\(L(\frac{7}{2})\) 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)$
bad3$C_1$ \( ( 1 - p^{2} T )^{2} \)
11$C_1$ \( ( 1 + p^{2} T )^{2} \)
good2$D_{4}$ \( 1 - 13 T + 49 p T^{2} - 13 p^{5} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 - 58 T + 6266 T^{2} - 58 p^{5} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 - 146 T + 7230 T^{2} - 146 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 10 p T + 701634 T^{2} + 10 p^{6} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 + 728 T + 1830542 T^{2} + 728 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 828 T - 865114 T^{2} + 828 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 238 T + 11988422 T^{2} + 238 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 24 p T + 22778374 T^{2} - 24 p^{6} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 + 10480 T + 63595902 T^{2} + 10480 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 1908 T + 74176718 T^{2} + 1908 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 - 36484 T + 564482438 T^{2} - 36484 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 9768 T + 237972854 T^{2} - 9768 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 - 43742 T + 935775830 T^{2} - 43742 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 12174 T + 457325722 T^{2} + 12174 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 2788 T + 1399448534 T^{2} + 2788 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 + 25302 T + 1815376826 T^{2} + 25302 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 40520 T + 1779236982 T^{2} + 40520 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 31386 T + 3698331094 T^{2} - 31386 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 46780 T + 4437463638 T^{2} + 46780 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 16850 T + 5148027246 T^{2} + 16850 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 79440 T + 5477266486 T^{2} - 79440 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 54204 T + 6019532470 T^{2} + 54204 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 241568 T + 30950947518 T^{2} + 241568 p^{5} T^{3} + p^{10} T^{4} \)
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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

−15.50465083808900918992923520188, −14.99325194156697193416206247668, −14.40524554217815183259477101427, −14.21788260677073977346304498900, −13.57594132238357617373735500931, −13.36877286391376741900515804397, −12.51282127289383014778852372457, −12.34007259110501297233343280900, −11.08628983909474475216419951988, −10.62439587140345802608846275568, −9.307421501695817151644839556853, −9.125902147853088621455115513635, −7.959609500874767055202244853026, −7.25040400667280707185793066363, −5.80663917185888314720906691634, −5.63390027193040633421602946195, −4.40693769843982712037763401810, −4.09020043256355535371557988808, −2.70176463886111205392393810501, −2.03685541957281458688932943194, 2.03685541957281458688932943194, 2.70176463886111205392393810501, 4.09020043256355535371557988808, 4.40693769843982712037763401810, 5.63390027193040633421602946195, 5.80663917185888314720906691634, 7.25040400667280707185793066363, 7.959609500874767055202244853026, 9.125902147853088621455115513635, 9.307421501695817151644839556853, 10.62439587140345802608846275568, 11.08628983909474475216419951988, 12.34007259110501297233343280900, 12.51282127289383014778852372457, 13.36877286391376741900515804397, 13.57594132238357617373735500931, 14.21788260677073977346304498900, 14.40524554217815183259477101427, 14.99325194156697193416206247668, 15.50465083808900918992923520188

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