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Properties

Label 2-882-7.5-c2-0-13
Degree $2$
Conductor $882$
Sign $0.832 - 0.553i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.999 − 1.73i)4-s + (7.24 + 4.18i)5-s − 2.82·8-s + (10.2 − 5.91i)10-s + (3 + 5.19i)11-s + 17.8i·13-s + (−2.00 + 3.46i)16-s + (−16.2 + 9.37i)17-s + (14.7 + 8.51i)19-s − 16.7i·20-s + 8.48·22-s + (6.72 − 11.6i)23-s + (22.4 + 38.9i)25-s + (21.8 + 12.6i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (1.44 + 0.836i)5-s − 0.353·8-s + (1.02 − 0.591i)10-s + (0.272 + 0.472i)11-s + 1.37i·13-s + (−0.125 + 0.216i)16-s + (−0.955 + 0.551i)17-s + (0.775 + 0.447i)19-s − 0.836i·20-s + 0.385·22-s + (0.292 − 0.506i)23-s + (0.898 + 1.55i)25-s + (0.841 + 0.485i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.669850337\)
\(L(\frac12)\) \(\approx\) \(2.669850337\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-7.24 - 4.18i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 17.8iT - 169T^{2} \)
17 \( 1 + (16.2 - 9.37i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.7 - 8.51i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-6.72 + 11.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 33.9T + 841T^{2} \)
31 \( 1 + (12.7 - 7.37i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (2.98 - 5.17i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 35.2iT - 1.68e3T^{2} \)
43 \( 1 - 15.4T + 1.84e3T^{2} \)
47 \( 1 + (28.7 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (17.2 + 29.9i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-23.6 + 13.6i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-34.9 - 20.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-57.1 - 99.0i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 18.6T + 5.04e3T^{2} \)
73 \( 1 + (-101. + 58.5i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-44.1 + 76.5i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 75.7iT - 6.88e3T^{2} \)
89 \( 1 + (18 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 30.5iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.900865126984285044446389180230, −9.557656775645400850261942594797, −8.673360246293950935582016928839, −7.09982448015459729271057825384, −6.50646523298619175836973026823, −5.67420337430965103043673963074, −4.61541498474185050625115214476, −3.50979265591099789635513051321, −2.24961268277246578405710696529, −1.66822693466611157427724257612, 0.76606488849368643904385243848, 2.25332645593227090224806523461, 3.50313052271167810548663863676, 4.94296737508998229011540241242, 5.45220163431749512709517449049, 6.16522410797379262053808125077, 7.21116730850747500706771789337, 8.188506803910121360122490975134, 9.186821389813148355065762532079, 9.464856363610213038090757908475

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