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Properties

Label 2-882-7.5-c2-0-2
Degree $2$
Conductor $882$
Sign $-0.832 + 0.553i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (2.74 + 1.58i)5-s + 2.82·8-s + (−3.87 + 2.23i)10-s + (−6.62 − 11.4i)11-s + 5.49i·13-s + (−2.00 + 3.46i)16-s + (−11.7 + 6.77i)17-s + (0.621 + 0.358i)19-s − 6.33i·20-s + 18.7·22-s + (−1.13 + 1.96i)23-s + (−7.48 − 12.9i)25-s + (−6.72 − 3.88i)26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.548 + 0.316i)5-s + 0.353·8-s + (−0.387 + 0.223i)10-s + (−0.601 − 1.04i)11-s + 0.422i·13-s + (−0.125 + 0.216i)16-s + (−0.690 + 0.398i)17-s + (0.0327 + 0.0188i)19-s − 0.316i·20-s + 0.851·22-s + (−0.0493 + 0.0855i)23-s + (−0.299 − 0.518i)25-s + (−0.258 − 0.149i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.832 + 0.553i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.832 + 0.553i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1049535721\)
\(L(\frac12)\) \(\approx\) \(0.1049535721\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-2.74 - 1.58i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.62 + 11.4i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 5.49iT - 169T^{2} \)
17 \( 1 + (11.7 - 6.77i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-0.621 - 0.358i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (1.13 - 1.96i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 20.4T + 841T^{2} \)
31 \( 1 + (21.3 - 12.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (32.4 - 56.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.0iT - 1.68e3T^{2} \)
43 \( 1 - 6.48T + 1.84e3T^{2} \)
47 \( 1 + (-41.3 - 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-11.0 - 19.0i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (72.5 - 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (57.3 + 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (46.3 + 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 48.4T + 5.04e3T^{2} \)
73 \( 1 + (113. - 65.4i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-38.1 + 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 107. iT - 6.88e3T^{2} \)
89 \( 1 + (145. + 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 25.5iT - 9.40e3T^{2} \)
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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

−10.48091925293849632542779785662, −9.439198272075778992923484485989, −8.744325252106182917841956767575, −7.946659530702521087617017833431, −6.97650105561894912647919070514, −6.14229048973151008689058510889, −5.49025276980308452893117260333, −4.31222031291706094432464058832, −2.99306947250500265075782239028, −1.68309290733084587553488219944, 0.03638163680107825820690484324, 1.68296562144759337011922319915, 2.55416207430979637727058779785, 3.88876131320758498567255797934, 4.94848447080990388650997581927, 5.76910362673495159519430944936, 7.13282959512998953991311057381, 7.74031701708941384141444472813, 8.939385409124011315948537319240, 9.417959918953400230538210273325

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