Properties

Label 2-882-63.5-c1-0-9
Degree $2$
Conductor $882$
Sign $0.488 - 0.872i$
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  + i·2-s + (−0.149 − 1.72i)3-s − 4-s + (0.948 + 1.64i)5-s + (1.72 − 0.149i)6-s i·8-s + (−2.95 + 0.514i)9-s + (−1.64 + 0.948i)10-s + (0.363 + 0.209i)11-s + (0.149 + 1.72i)12-s + (1.54 + 0.891i)13-s + (2.69 − 1.88i)15-s + 16-s + (1.05 + 1.83i)17-s + (−0.514 − 2.95i)18-s + (5.50 + 3.17i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.0861 − 0.996i)3-s − 0.5·4-s + (0.424 + 0.735i)5-s + (0.704 − 0.0608i)6-s − 0.353i·8-s + (−0.985 + 0.171i)9-s + (−0.519 + 0.300i)10-s + (0.109 + 0.0631i)11-s + (0.0430 + 0.498i)12-s + (0.428 + 0.247i)13-s + (0.695 − 0.486i)15-s + 0.250·16-s + (0.256 + 0.444i)17-s + (−0.121 − 0.696i)18-s + (1.26 + 0.728i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28340 + 0.752179i\)
\(L(\frac12)\) \(\approx\) \(1.28340 + 0.752179i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.149 + 1.72i)T \)
7 \( 1 \)
good5 \( 1 + (-0.948 - 1.64i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.363 - 0.209i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.54 - 0.891i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.05 - 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.50 - 3.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.49 - 3.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.57 + 2.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 + (-5.29 + 9.16i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.36 - 4.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.03 - 6.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.62T + 47T^{2} \)
53 \( 1 + (6.52 - 3.76i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.10T + 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 1.21T + 67T^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 + (9.31 - 5.37i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 3.61T + 79T^{2} \)
83 \( 1 + (-5.98 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.56 - 2.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 + 5.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.16954839705143047193582442477, −9.367244636424858852989456062716, −8.241841982927115180174655291794, −7.68012930869410766852913340057, −6.79528414410664938074165903082, −6.08507978701522531563759785351, −5.48677613097309692218933959210, −3.93587363259681103018319337982, −2.71743939771737641829981676885, −1.33699225275192674511299117458, 0.837725651522135540398372588272, 2.53240275312349301219945267879, 3.60110543969476657821571421852, 4.60968096609214079469217241617, 5.31281205968046129028203203660, 6.18785452185565118688910606406, 7.74123522367679266191335102863, 8.753089284338080509574632349310, 9.255411588431521995628771695056, 10.05202669992873030192404088012

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