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Properties

Label 2-882-63.5-c1-0-0
Degree $2$
Conductor $882$
Sign $-0.0734 - 0.997i$
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  − i·2-s + (−1.60 + 0.639i)3-s − 4-s + (−1.35 − 2.33i)5-s + (0.639 + 1.60i)6-s + i·8-s + (2.18 − 2.05i)9-s + (−2.33 + 1.35i)10-s + (0.205 + 0.118i)11-s + (1.60 − 0.639i)12-s + (−2.31 − 1.33i)13-s + (3.66 + 2.90i)15-s + 16-s + (−2.93 − 5.08i)17-s + (−2.05 − 2.18i)18-s + (0.998 + 0.576i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.929 + 0.369i)3-s − 0.5·4-s + (−0.603 − 1.04i)5-s + (0.261 + 0.657i)6-s + 0.353i·8-s + (0.727 − 0.686i)9-s + (−0.739 + 0.426i)10-s + (0.0621 + 0.0358i)11-s + (0.464 − 0.184i)12-s + (−0.641 − 0.370i)13-s + (0.947 + 0.748i)15-s + 0.250·16-s + (−0.711 − 1.23i)17-s + (−0.485 − 0.514i)18-s + (0.229 + 0.132i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0734 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0734 - 0.997i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.0734 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0366472 + 0.0394436i\)
\(L(\frac12)\) \(\approx\) \(0.0366472 + 0.0394436i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.60 - 0.639i)T \)
7 \( 1 \)
good5 \( 1 + (1.35 + 2.33i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.205 - 0.118i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.31 + 1.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.93 + 5.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.998 - 0.576i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.02 - 4.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.88 + 5.13i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.02iT - 31T^{2} \)
37 \( 1 + (4.84 - 8.38i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.81 - 6.61i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.69 - 4.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.442T + 47T^{2} \)
53 \( 1 + (0.189 - 0.109i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.96T + 59T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 + 8.96T + 67T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 + (6.28 - 3.62i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 + (0.762 + 1.32i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.76 - 11.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.37 + 0.795i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.21015142114871542988774800650, −9.839549114288356791298914186239, −8.833463587759954595513327651359, −8.002969810595301392978232537149, −6.90391819972686124282357486017, −5.71013099637839197450738511035, −4.72275905059588088613349205790, −4.40375367576252574713338382457, −3.01164130409644519321479506528, −1.24019892749295061597860180818, 0.03220087048538205498148839297, 2.14916389784397352216985107040, 3.79199883679285726432252914402, 4.64860523796423654206410559290, 5.82867960243096333902155226392, 6.56963942168447781865194782368, 7.14808353647919719959496691755, 7.925331759073264549971885458626, 8.883322665762761199021885005167, 10.30341458698503415615289723499

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