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Properties

Label 2-882-63.38-c1-0-27
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: $\chi_{882} (227, \cdot )$
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

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

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