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Properties

Label 2-882-63.25-c1-0-9
Degree $2$
Conductor $882$
Sign $0.987 - 0.159i$
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  − 2-s + (−1.71 − 0.272i)3-s + 4-s + (−1.59 + 2.75i)5-s + (1.71 + 0.272i)6-s − 8-s + (2.85 + 0.931i)9-s + (1.59 − 2.75i)10-s + (−1.59 − 2.75i)11-s + (−1.71 − 0.272i)12-s + (−2.85 − 4.93i)13-s + (3.47 − 4.28i)15-s + 16-s + (0.760 − 1.31i)17-s + (−2.85 − 0.931i)18-s + (0.641 + 1.11i)19-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.987 − 0.157i)3-s + 0.5·4-s + (−0.711 + 1.23i)5-s + (0.698 + 0.111i)6-s − 0.353·8-s + (0.950 + 0.310i)9-s + (0.503 − 0.871i)10-s + (−0.479 − 0.830i)11-s + (−0.493 − 0.0785i)12-s + (−0.790 − 1.36i)13-s + (0.896 − 1.10i)15-s + 0.250·16-s + (0.184 − 0.319i)17-s + (−0.672 − 0.219i)18-s + (0.147 + 0.254i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.987 - 0.159i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (655, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.548154 + 0.0440109i\)
\(L(\frac12)\) \(\approx\) \(0.548154 + 0.0440109i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.71 + 0.272i)T \)
7 \( 1 \)
good5 \( 1 + (1.59 - 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.85 + 4.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.760 + 1.31i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.641 - 1.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.11 - 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.42T + 31T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.82T + 47T^{2} \)
53 \( 1 + (-1.02 + 1.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + 3.12T + 61T^{2} \)
67 \( 1 - 10.9T + 67T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (-2.48 + 4.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 4.13T + 79T^{2} \)
83 \( 1 + (-4.03 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.112 + 0.195i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.42 - 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.49796708546770551680794532077, −9.598783469884134865579822265765, −8.130320050414514327583450858768, −7.61280791611295906753179627175, −6.89591090708176163256588153255, −5.95927779163371304003709253295, −5.11733528132779011896108262957, −3.54913091565098640196367257050, −2.61823683761933418785395393395, −0.65952367629469237131981563818, 0.69689977337598304238450173717, 2.11904193427939201378241716438, 4.23687639745760860440195867457, 4.63224159373872364084689957492, 5.74949606970665667459024150974, 6.84563409625631462453437434694, 7.59513372424388163774831136057, 8.446865224150766477764292812088, 9.504701628896832306200027584939, 9.874674984902908969080544945258

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