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Properties

Label 2-882-63.16-c1-0-24
Degree $2$
Conductor $882$
Sign $-0.159 - 0.987i$
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 + (1.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + 4.37·5-s + (−0.5 + 1.65i)6-s − 0.999·8-s + (−0.186 + 2.99i)9-s + (2.18 + 3.78i)10-s − 1.37·11-s + (−1.68 + 0.396i)12-s + (−1 − 1.73i)13-s + (5.18 + 5.51i)15-s + (−0.5 − 0.866i)16-s + (−0.686 − 1.18i)17-s + (−2.68 + 1.33i)18-s + (−2.5 + 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.684 + 0.728i)3-s + (−0.249 + 0.433i)4-s + 1.95·5-s + (−0.204 + 0.677i)6-s − 0.353·8-s + (−0.0620 + 0.998i)9-s + (0.691 + 1.19i)10-s − 0.413·11-s + (−0.486 + 0.114i)12-s + (−0.277 − 0.480i)13-s + (1.33 + 1.42i)15-s + (−0.125 − 0.216i)16-s + (−0.166 − 0.288i)17-s + (−0.633 + 0.314i)18-s + (−0.573 + 0.993i)19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.159 - 0.987i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94200 + 2.28105i\)
\(L(\frac12)\) \(\approx\) \(1.94200 + 2.28105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 \)
good5 \( 1 - 4.37T + 5T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.686 + 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.62T + 23T^{2} \)
29 \( 1 + (-4.37 + 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.31 + 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.37 - 7.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.05 + 8.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.55 + 2.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.11T + 71T^{2} \)
73 \( 1 + (6.05 + 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.55 - 4.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.74 - 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.05 + 7.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.15388273758905138469011438279, −9.573869908597193946939798934489, −8.729206022098523431224086700170, −7.974530701591296652592662114682, −6.79131042432532082199724315142, −5.79079032941488174358061833422, −5.28861207707756843996798095166, −4.23745593571510031171132037109, −2.88134115319943188416323233101, −2.04877909036969525082440318885, 1.36223859627416445933931523346, 2.27914554741916594442797353271, 2.93956834741724938854186651480, 4.53408477818591859292982505535, 5.58281611318310938060718697071, 6.40302877460966493467995695972, 7.10955387652982794376267245686, 8.610668971720438054357727869634, 9.087927441095640285887805432955, 9.935291805904967166325077689962

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