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Properties

Label 2-882-441.131-c1-0-54
Degree $2$
Conductor $882$
Sign $0.191 + 0.981i$
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  + (0.781 + 0.623i)2-s + (1.06 − 1.36i)3-s + (0.222 + 0.974i)4-s + (−0.278 − 3.72i)5-s + (1.68 − 0.405i)6-s + (2.25 − 1.37i)7-s + (−0.433 + 0.900i)8-s + (−0.735 − 2.90i)9-s + (2.10 − 3.08i)10-s + (2.85 − 1.11i)11-s + (1.56 + 0.733i)12-s + (−3.80 + 1.49i)13-s + (2.62 + 0.330i)14-s + (−5.38 − 3.57i)15-s + (−0.900 + 0.433i)16-s + (−5.72 + 1.76i)17-s + ⋯
L(s)  = 1  + (0.552 + 0.440i)2-s + (0.614 − 0.789i)3-s + (0.111 + 0.487i)4-s + (−0.124 − 1.66i)5-s + (0.687 − 0.165i)6-s + (0.853 − 0.520i)7-s + (−0.153 + 0.318i)8-s + (−0.245 − 0.969i)9-s + (0.664 − 0.974i)10-s + (0.859 − 0.337i)11-s + (0.452 + 0.211i)12-s + (−1.05 + 0.414i)13-s + (0.701 + 0.0883i)14-s + (−1.38 − 0.923i)15-s + (−0.225 + 0.108i)16-s + (−1.38 + 0.428i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.191 + 0.981i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01117 - 1.65586i\)
\(L(\frac12)\) \(\approx\) \(2.01117 - 1.65586i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 - 0.623i)T \)
3 \( 1 + (-1.06 + 1.36i)T \)
7 \( 1 + (-2.25 + 1.37i)T \)
good5 \( 1 + (0.278 + 3.72i)T + (-4.94 + 0.745i)T^{2} \)
11 \( 1 + (-2.85 + 1.11i)T + (8.06 - 7.48i)T^{2} \)
13 \( 1 + (3.80 - 1.49i)T + (9.52 - 8.84i)T^{2} \)
17 \( 1 + (5.72 - 1.76i)T + (14.0 - 9.57i)T^{2} \)
19 \( 1 + (0.0199 + 0.0115i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.13 - 4.45i)T + (-1.71 + 22.9i)T^{2} \)
29 \( 1 + (0.857 + 2.78i)T + (-23.9 + 16.3i)T^{2} \)
31 \( 1 - 6.03iT - 31T^{2} \)
37 \( 1 + (-3.39 - 3.14i)T + (2.76 + 36.8i)T^{2} \)
41 \( 1 + (-6.46 + 4.40i)T + (14.9 - 38.1i)T^{2} \)
43 \( 1 + (-7.27 - 4.96i)T + (15.7 + 40.0i)T^{2} \)
47 \( 1 + (-4.70 + 5.90i)T + (-10.4 - 45.8i)T^{2} \)
53 \( 1 + (0.841 + 0.907i)T + (-3.96 + 52.8i)T^{2} \)
59 \( 1 + (-12.8 + 6.19i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (2.83 + 0.646i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 - 5.13T + 67T^{2} \)
71 \( 1 + (-0.0977 + 0.0223i)T + (63.9 - 30.8i)T^{2} \)
73 \( 1 + (9.71 + 3.81i)T + (53.5 + 49.6i)T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + (4.00 - 10.1i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (-1.07 + 0.162i)T + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (9.61 - 5.55i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.448855152866305698689957283651, −8.896920854248884899171379497007, −8.236988160042266211856830923031, −7.42937950496645102953270253478, −6.65024182044164756107795859514, −5.44612266178134752885504790816, −4.54706300481105601082344461483, −3.88091545040778709295773515756, −2.15257479437154566119436741646, −1.02598087814531639650723876892, 2.42021454797951079831901568047, 2.61768780353195770966749934846, 4.01369971310581825631692387360, 4.67079749313896633002988916855, 5.83787892484601308123095208736, 6.96083482809229392078569170737, 7.64580081913747734666767011628, 8.918928747356490716148451803572, 9.575358348403594240517639233327, 10.56049411670596589316392496780

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