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Properties

Label 2-882-63.41-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.460 - 0.887i$
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  + (0.866 − 0.5i)2-s + (0.0537 − 1.73i)3-s + (0.499 − 0.866i)4-s + (−1.99 + 3.45i)5-s + (−0.819 − 1.52i)6-s − 0.999i·8-s + (−2.99 − 0.186i)9-s + 3.98i·10-s + (−1.43 + 0.828i)11-s + (−1.47 − 0.912i)12-s + (−2.60 − 1.50i)13-s + (5.87 + 3.63i)15-s + (−0.5 − 0.866i)16-s − 7.45·17-s + (−2.68 + 1.33i)18-s + 4.57i·19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0310 − 0.999i)3-s + (0.249 − 0.433i)4-s + (−0.891 + 1.54i)5-s + (−0.334 − 0.623i)6-s − 0.353i·8-s + (−0.998 − 0.0620i)9-s + 1.26i·10-s + (−0.432 + 0.249i)11-s + (−0.425 − 0.263i)12-s + (−0.723 − 0.417i)13-s + (1.51 + 0.938i)15-s + (−0.125 − 0.216i)16-s − 1.80·17-s + (−0.633 + 0.314i)18-s + 1.04i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.460 - 0.887i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160817 + 0.264692i\)
\(L(\frac12)\) \(\approx\) \(0.160817 + 0.264692i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.0537 + 1.73i)T \)
7 \( 1 \)
good5 \( 1 + (1.99 - 3.45i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.43 - 0.828i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.60 + 1.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 7.45T + 17T^{2} \)
19 \( 1 - 4.57iT - 19T^{2} \)
23 \( 1 + (0.253 + 0.146i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.18 - 1.83i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.08 - 2.35i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.01T + 37T^{2} \)
41 \( 1 + (4.88 - 8.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.29 + 2.25i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.80 + 11.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.916iT - 53T^{2} \)
59 \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.75 + 3.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.02 - 3.51i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.9iT - 71T^{2} \)
73 \( 1 + 4.44iT - 73T^{2} \)
79 \( 1 + (1.42 + 2.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.59 - 4.49i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 + (2.51 - 1.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.65120520997865695036050177148, −9.914918718498828720920622097690, −8.393387816483935092141839289246, −7.68460474737948536836679774688, −6.82506813314561091231807314400, −6.42253889365781247598595522342, −5.12323114517901959955509101965, −3.84775482245996623242731007177, −2.87759525224616532321409904815, −2.11859688807033864457259357028, 0.10996427945163915984128165142, 2.51150872664694529106012055122, 3.90530598497565240816689906146, 4.64984219436682007964637499953, 4.96032547197147888161999553855, 6.15534087929620536091864998230, 7.41296323746400869059643555424, 8.322480549843941979482565333994, 8.943377248396841077218933451851, 9.595324644644318909191766650655

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