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Properties

Label 2-882-441.101-c1-0-30
Degree $2$
Conductor $882$
Sign $0.125 + 0.992i$
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 + (−1.53 + 0.810i)3-s + (0.222 − 0.974i)4-s + (−0.265 + 3.54i)5-s + (0.691 − 1.58i)6-s + (0.560 − 2.58i)7-s + (0.433 + 0.900i)8-s + (1.68 − 2.48i)9-s + (−2.00 − 2.93i)10-s + (2.48 + 0.976i)11-s + (0.449 + 1.67i)12-s + (−2.80 − 1.09i)13-s + (1.17 + 2.37i)14-s + (−2.46 − 5.64i)15-s + (−0.900 − 0.433i)16-s + (−4.22 − 1.30i)17-s + ⋯
L(s)  = 1  + (−0.552 + 0.440i)2-s + (−0.883 + 0.467i)3-s + (0.111 − 0.487i)4-s + (−0.118 + 1.58i)5-s + (0.282 − 0.648i)6-s + (0.211 − 0.977i)7-s + (0.153 + 0.318i)8-s + (0.562 − 0.826i)9-s + (−0.633 − 0.929i)10-s + (0.750 + 0.294i)11-s + (0.129 + 0.482i)12-s + (−0.776 − 0.304i)13-s + (0.313 + 0.633i)14-s + (−0.636 − 1.45i)15-s + (−0.225 − 0.108i)16-s + (−1.02 − 0.316i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.125 + 0.992i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.125 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102399 - 0.0902334i\)
\(L(\frac12)\) \(\approx\) \(0.102399 - 0.0902334i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 - 0.810i)T \)
7 \( 1 + (-0.560 + 2.58i)T \)
good5 \( 1 + (0.265 - 3.54i)T + (-4.94 - 0.745i)T^{2} \)
11 \( 1 + (-2.48 - 0.976i)T + (8.06 + 7.48i)T^{2} \)
13 \( 1 + (2.80 + 1.09i)T + (9.52 + 8.84i)T^{2} \)
17 \( 1 + (4.22 + 1.30i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (5.15 - 2.97i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.24 - 6.73i)T + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (-3.00 + 9.75i)T + (-23.9 - 16.3i)T^{2} \)
31 \( 1 + 5.52iT - 31T^{2} \)
37 \( 1 + (-6.05 + 5.61i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-3.00 - 2.04i)T + (14.9 + 38.1i)T^{2} \)
43 \( 1 + (1.72 - 1.17i)T + (15.7 - 40.0i)T^{2} \)
47 \( 1 + (2.97 + 3.72i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-1.66 + 1.79i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (10.7 + 5.18i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-0.0189 + 0.00432i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 - 3.04T + 67T^{2} \)
71 \( 1 + (2.15 + 0.491i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.79 - 1.88i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 - 7.65T + 79T^{2} \)
83 \( 1 + (4.15 + 10.5i)T + (-60.8 + 56.4i)T^{2} \)
89 \( 1 + (-5.84 - 0.881i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + (5.24 + 3.02i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.954406730982375767369399708517, −9.558334353798299115602080668700, −7.898580948214785324572156116171, −7.34198114406266589288650925783, −6.43536653714853531348751679912, −6.04578360754804979460381198593, −4.46647026520590422908226485167, −3.81565512819713454451139598168, −2.12328911202370102730777340833, −0.090601304422503543326881358105, 1.34885197658482956094549753057, 2.34490946813997085582239520189, 4.44996737232062067937795899244, 4.80793151984896358977527665072, 6.08498960518495931158214597590, 6.80627696303338236557695351740, 8.186733334331241046087216295432, 8.681024685422258665712702164154, 9.273505450618219554414548112055, 10.44104661741907992829186606171

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