Properties

Label 2-882-63.4-c1-0-35
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: $\chi_{882} (67, \cdot )$
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

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

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