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Properties

Label 2-882-147.101-c1-0-12
Degree $2$
Conductor $882$
Sign $0.910 + 0.412i$
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  + (0.930 + 0.365i)2-s + (0.733 + 0.680i)4-s + (−3.09 − 2.10i)5-s + (−0.605 + 2.57i)7-s + (0.433 + 0.900i)8-s + (−2.10 − 3.09i)10-s + (−0.198 − 1.31i)11-s + (5.18 − 4.13i)13-s + (−1.50 + 2.17i)14-s + (0.0747 + 0.997i)16-s + (3.42 + 1.05i)17-s + (6.36 − 3.67i)19-s + (−0.832 − 3.64i)20-s + (0.295 − 1.29i)22-s + (−1.67 − 5.41i)23-s + ⋯
L(s)  = 1  + (0.658 + 0.258i)2-s + (0.366 + 0.340i)4-s + (−1.38 − 0.942i)5-s + (−0.228 + 0.973i)7-s + (0.153 + 0.318i)8-s + (−0.666 − 0.977i)10-s + (−0.0597 − 0.396i)11-s + (1.43 − 1.14i)13-s + (−0.402 + 0.581i)14-s + (0.0186 + 0.249i)16-s + (0.831 + 0.256i)17-s + (1.46 − 0.843i)19-s + (−0.186 − 0.815i)20-s + (0.0630 − 0.276i)22-s + (−0.348 − 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82570 - 0.394451i\)
\(L(\frac12)\) \(\approx\) \(1.82570 - 0.394451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 \)
7 \( 1 + (0.605 - 2.57i)T \)
good5 \( 1 + (3.09 + 2.10i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (0.198 + 1.31i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-5.18 + 4.13i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (-3.42 - 1.05i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-6.36 + 3.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.67 + 5.41i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-6.48 + 1.48i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.438 + 0.253i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.93 - 1.79i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (6.06 - 2.91i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (0.521 + 0.251i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.91 + 7.42i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (5.04 - 5.43i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-10.7 + 7.29i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (2.41 + 2.60i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (4.52 - 7.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.14 + 1.17i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (-5.00 + 1.96i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-0.310 - 0.537i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.06 + 3.84i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (5.64 + 0.850i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 16.8iT - 97T^{2} \)
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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

−10.16376441208313132941161598645, −8.795131438888414746970860224396, −8.369752936404157619978970458075, −7.72738184171065828248024980309, −6.47141144473951570385842520741, −5.52493253332119448127708862217, −4.86430852181342258430173624345, −3.65263504108646272741255470612, −3.01400093352310418104811658121, −0.875383269920944053517399389711, 1.32787593933197990729598733818, 3.30145959263332828833199427403, 3.62214375185178415751066708539, 4.51714565375175547292181930777, 5.88347103771814218830168586975, 6.94000173437482250525455850156, 7.37746651508212363602223537947, 8.257175192176388369309186095976, 9.649856699234028178273312798710, 10.40549092103188817219905144050

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