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Properties

Label 2-882-9.7-c1-0-11
Degree $2$
Conductor $882$
Sign $-0.528 - 0.848i$
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  + (−0.5 + 0.866i)2-s + (1.71 + 0.272i)3-s + (−0.499 − 0.866i)4-s + (0.880 + 1.52i)5-s + (−1.09 + 1.34i)6-s + 0.999·8-s + (2.85 + 0.931i)9-s − 1.76·10-s + (−3.06 + 5.30i)11-s + (−0.619 − 1.61i)12-s + (0.380 + 0.658i)13-s + (1.09 + 2.84i)15-s + (−0.5 + 0.866i)16-s − 6.84·17-s + (−2.23 + 2.00i)18-s + 1.94·19-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.987 + 0.157i)3-s + (−0.249 − 0.433i)4-s + (0.393 + 0.681i)5-s + (−0.445 + 0.549i)6-s + 0.353·8-s + (0.950 + 0.310i)9-s − 0.556·10-s + (−0.923 + 1.59i)11-s + (−0.178 − 0.466i)12-s + (0.105 + 0.182i)13-s + (0.281 + 0.735i)15-s + (−0.125 + 0.216i)16-s − 1.65·17-s + (−0.526 + 0.472i)18-s + 0.445·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.528 - 0.848i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.528 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.831548 + 1.49764i\)
\(L(\frac12)\) \(\approx\) \(0.831548 + 1.49764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (-1.71 - 0.272i)T \)
7 \( 1 \)
good5 \( 1 + (-0.880 - 1.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3.06 - 5.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.380 - 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + (-0.210 - 0.364i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.732 + 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 + (-3.47 - 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.830 + 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.225T + 53T^{2} \)
59 \( 1 + (-0.993 - 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 0.306T + 73T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.56 + 2.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 2.60T + 89T^{2} \)
97 \( 1 + (-1.81 + 3.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.31397973812710590349614598632, −9.459670611547965671667911541122, −8.784781945344793136144349658884, −7.81976495020580836782944962897, −7.09112480429062792140826746670, −6.49896535309886962465105494599, −5.02084635469026355603953107960, −4.29106550015204442582309994485, −2.76572925270284119181190736177, −1.94983388478908225270867261004, 0.823122546367163808942760802401, 2.24212920294888074247423571400, 3.09647240771912127959814468508, 4.20865674559227692349734677090, 5.31781205151041342751254757703, 6.48258607029176514653323892950, 7.73061661372739990094039835447, 8.383665634469145396393673983964, 9.008585592590557317805555386991, 9.617017308396910116800236544235

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