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Properties

Label 2-882-441.130-c1-0-14
Degree $2$
Conductor $882$
Sign $0.983 - 0.180i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 − 0.563i)2-s + (−1.62 + 0.593i)3-s + (0.365 − 0.930i)4-s + (0.152 + 0.669i)5-s + (−1.01 + 1.40i)6-s + (−2.64 + 0.0878i)7-s + (−0.222 − 0.974i)8-s + (2.29 − 1.93i)9-s + (0.503 + 0.466i)10-s + (2.90 + 1.40i)11-s + (−0.0421 + 1.73i)12-s + (0.00890 − 0.00607i)13-s + (−2.13 + 1.56i)14-s + (−0.645 − 0.998i)15-s + (−0.733 − 0.680i)16-s + (−0.386 − 0.985i)17-s + ⋯
L(s)  = 1  + (0.584 − 0.398i)2-s + (−0.939 + 0.342i)3-s + (0.182 − 0.465i)4-s + (0.0683 + 0.299i)5-s + (−0.412 + 0.574i)6-s + (−0.999 + 0.0332i)7-s + (−0.0786 − 0.344i)8-s + (0.765 − 0.643i)9-s + (0.159 + 0.147i)10-s + (0.877 + 0.422i)11-s + (−0.0121 + 0.499i)12-s + (0.00246 − 0.00168i)13-s + (−0.570 + 0.417i)14-s + (−0.166 − 0.257i)15-s + (−0.183 − 0.170i)16-s + (−0.0937 − 0.238i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.983 - 0.180i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.983 - 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51045 + 0.137570i\)
\(L(\frac12)\) \(\approx\) \(1.51045 + 0.137570i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 + 0.563i)T \)
3 \( 1 + (1.62 - 0.593i)T \)
7 \( 1 + (2.64 - 0.0878i)T \)
good5 \( 1 + (-0.152 - 0.669i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (-2.90 - 1.40i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-0.00890 + 0.00607i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.386 + 0.985i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (-0.471 - 0.817i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.34 - 5.44i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-4.74 - 0.715i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-4.01 - 6.95i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.90 - 0.287i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (1.83 + 0.566i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-1.22 + 0.377i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-1.06 + 0.723i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (0.155 - 0.0233i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (-11.9 + 3.67i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (1.50 + 3.84i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-3.59 - 6.23i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.09 + 1.37i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (0.878 - 11.7i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (1.20 - 2.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.41 + 3.01i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (11.0 + 7.52i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (1.03 + 1.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.17928756148111209118093198870, −9.713202163451699254826164097509, −8.791319673053114948626949887240, −6.95604039915331806177493345733, −6.74738827123141575085193398421, −5.71785633401630524667276366936, −4.82488615033196965371032544605, −3.83992082167509081982551022319, −2.91347325384302368662289460918, −1.14056940478222276781190315547, 0.862836691489045393712111077804, 2.70561204477913874831141447794, 4.01501693923790579009994491330, 4.88007098850262127975687819912, 5.94040847382147780407643490973, 6.51608294607747482845311702669, 7.13463799718861487302743681728, 8.347779473645098380927333717433, 9.243043194706389607125235727825, 10.23317613273607440774111723008

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