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Properties

Label 2-882-9.4-c1-0-12
Degree $2$
Conductor $882$
Sign $-0.173 - 0.984i$
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 + (−1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1 − 1.73i)5-s + (−1.5 − 0.866i)6-s − 0.999·8-s + (1.5 − 2.59i)9-s + 1.99·10-s + (−0.5 − 0.866i)11-s − 1.73i·12-s + (−3 + 5.19i)13-s + 3.46i·15-s + (−0.5 − 0.866i)16-s + 5·17-s + 3·18-s + 7·19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.447 − 0.774i)5-s + (−0.612 − 0.353i)6-s − 0.353·8-s + (0.5 − 0.866i)9-s + 0.632·10-s + (−0.150 − 0.261i)11-s − 0.499i·12-s + (−0.832 + 1.44i)13-s + 0.894i·15-s + (−0.125 − 0.216i)16-s + 1.21·17-s + 0.707·18-s + 1.60·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.897944 + 1.07012i\)
\(L(\frac12)\) \(\approx\) \(0.897944 + 1.07012i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (3.5 - 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8 - 13.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.17318625049513387729053349998, −9.464368134966080869921280332045, −8.923161009139398211223185384696, −7.55062949446040239730601667764, −6.89554135753452666175233274720, −5.65840348992276005706689190969, −5.31037999706034614982556854004, −4.43063544192618199994978382334, −3.31662964412352303432993514544, −1.30612581939876014499866481038, 0.76780581499665480973680466339, 2.31378665176726382421743000014, 3.19622936436499456904140047322, 4.71388967540694120220463702532, 5.58581880094825867675902217517, 6.14317622676315631435875322403, 7.40588436511133023450500150930, 7.84107209090897571194132140503, 9.527895429530671893718249688983, 10.34416851717288209871023180369

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