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Properties

Label 2-882-7.3-c2-0-26
Degree $2$
Conductor $882$
Sign $-0.414 + 0.909i$
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 + (1.31 − 0.760i)5-s + 2.82·8-s + (−1.86 − 1.07i)10-s + (6.86 − 11.8i)11-s − 17.9i·13-s + (−2.00 − 3.46i)16-s + (22.3 + 12.9i)17-s + (−16.4 + 9.50i)19-s + 3.04i·20-s − 19.4·22-s + (9.67 + 16.7i)23-s + (−11.3 + 19.6i)25-s + (−21.9 + 12.6i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.263 − 0.152i)5-s + 0.353·8-s + (−0.186 − 0.107i)10-s + (0.623 − 1.08i)11-s − 1.38i·13-s + (−0.125 − 0.216i)16-s + (1.31 + 0.759i)17-s + (−0.866 + 0.500i)19-s + 0.152i·20-s − 0.882·22-s + (0.420 + 0.728i)23-s + (−0.453 + 0.785i)25-s + (−0.845 + 0.488i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.414 + 0.909i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.414 + 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.462572157\)
\(L(\frac12)\) \(\approx\) \(1.462572157\)
\(L(2)\) 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 \( 1 + (-1.31 + 0.760i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.86 + 11.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 17.9iT - 169T^{2} \)
17 \( 1 + (-22.3 - 12.9i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (16.4 - 9.50i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.67 - 16.7i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 10.5T + 841T^{2} \)
31 \( 1 + (13.0 + 7.54i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (29.2 + 50.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 54.0iT - 1.68e3T^{2} \)
43 \( 1 - 73.8T + 1.84e3T^{2} \)
47 \( 1 + (0.942 - 0.544i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-38.5 + 66.7i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (59.3 + 34.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (61.6 - 35.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-59.1 + 102. i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 11.0T + 5.04e3T^{2} \)
73 \( 1 + (29.6 + 17.1i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-30.8 - 53.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 97.0iT - 6.88e3T^{2} \)
89 \( 1 + (0.0207 - 0.0119i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 20.1iT - 9.40e3T^{2} \)
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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

−9.705414276051847736408662182726, −8.910277420758957532542147913123, −8.140418953767659422099088458809, −7.38313494619493175366902624300, −5.92050494248863876698769168311, −5.47241983252974790037005926278, −3.83830320610558289379701520206, −3.25401900972062153141207547483, −1.77604864852243129520195943940, −0.60056103535663382701043326776, 1.28837351957668689780884604037, 2.55326948288497370607940686781, 4.19931654164281220382235401277, 4.86756583199074742201185061791, 6.16098593781572420784398651981, 6.80164983542540648466330575147, 7.51068169209569518003777318788, 8.606033250438740347396067321479, 9.365960811876847450578011883618, 9.943295959174090970861775803059

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