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Properties

Label 2-882-63.41-c1-0-6
Degree $2$
Conductor $882$
Sign $-0.604 - 0.796i$
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 + (−1.59 + 0.681i)3-s + (0.499 − 0.866i)4-s + (0.714 − 1.23i)5-s + (1.03 − 1.38i)6-s + 0.999i·8-s + (2.07 − 2.17i)9-s + 1.42i·10-s + (−2.96 + 1.70i)11-s + (−0.206 + 1.71i)12-s + (−5.48 − 3.16i)13-s + (−0.294 + 2.45i)15-s + (−0.5 − 0.866i)16-s + 2.28·17-s + (−0.708 + 2.91i)18-s + 2.16i·19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.919 + 0.393i)3-s + (0.249 − 0.433i)4-s + (0.319 − 0.553i)5-s + (0.423 − 0.565i)6-s + 0.353i·8-s + (0.690 − 0.723i)9-s + 0.452i·10-s + (−0.892 + 0.515i)11-s + (−0.0594 + 0.496i)12-s + (−1.52 − 0.878i)13-s + (−0.0760 + 0.634i)15-s + (−0.125 − 0.216i)16-s + 0.553·17-s + (−0.167 + 0.687i)18-s + 0.497i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.604 - 0.796i$
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: \((2,\ 882,\ (\ :1/2),\ -0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202554 + 0.408285i\)
\(L(\frac12)\) \(\approx\) \(0.202554 + 0.408285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.681i)T \)
7 \( 1 \)
good5 \( 1 + (-0.714 + 1.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.96 - 1.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.48 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.28T + 17T^{2} \)
19 \( 1 - 2.16iT - 19T^{2} \)
23 \( 1 + (-6.97 - 4.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.298 - 0.172i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.76 + 2.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.15T + 37T^{2} \)
41 \( 1 + (-0.202 + 0.350i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.90 - 5.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.75 - 4.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.88iT - 53T^{2} \)
59 \( 1 + (5.51 - 9.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.94 - 5.73i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.12 - 3.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 + 0.232iT - 73T^{2} \)
79 \( 1 + (7.28 + 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.811 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.05T + 89T^{2} \)
97 \( 1 + (9.18 - 5.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.34871148310064889093602966562, −9.615675090694100479406541863900, −9.038876458532018416306005944037, −7.51238654602939743656630036028, −7.39493879112491843179038044899, −5.86600029015282536766762349455, −5.31666472563062797742971933427, −4.62906888962514003940095225434, −2.91058319774549338326025989796, −1.24178001823539971221064003760, 0.32414679962547293348455438541, 2.02479120630721269171637345733, 2.96665827163888454228085647208, 4.66769677801671423639585379638, 5.44719943822516587641301313238, 6.72343994702110512413754762763, 7.09791790239459865677853933468, 8.068837991130752165964419840985, 9.179637258118412274082754266452, 10.04856327053128635171279657153

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