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Properties

Label 2-882-63.38-c1-0-3
Degree $2$
Conductor $882$
Sign $-0.999 + 0.0139i$
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 + (0.307 + 1.70i)3-s − 4-s + (−1.55 + 2.68i)5-s + (1.70 − 0.307i)6-s + i·8-s + (−2.81 + 1.04i)9-s + (2.68 + 1.55i)10-s + (−4.10 + 2.37i)11-s + (−0.307 − 1.70i)12-s + (2.94 − 1.69i)13-s + (−5.05 − 1.81i)15-s + 16-s + (2.51 − 4.36i)17-s + (1.04 + 2.81i)18-s + (−2.29 + 1.32i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.177 + 0.984i)3-s − 0.5·4-s + (−0.693 + 1.20i)5-s + (0.695 − 0.125i)6-s + 0.353i·8-s + (−0.936 + 0.349i)9-s + (0.849 + 0.490i)10-s + (−1.23 + 0.715i)11-s + (−0.0887 − 0.492i)12-s + (0.815 − 0.471i)13-s + (−1.30 − 0.469i)15-s + 0.250·16-s + (0.610 − 1.05i)17-s + (0.247 + 0.662i)18-s + (−0.525 + 0.303i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 + 0.0139i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.999 + 0.0139i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00266747 - 0.383496i\)
\(L(\frac12)\) \(\approx\) \(0.00266747 - 0.383496i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.307 - 1.70i)T \)
7 \( 1 \)
good5 \( 1 + (1.55 - 2.68i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.10 - 2.37i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 + 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.51 + 4.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.29 - 1.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.99 + 1.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.92 + 5.15i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.97iT - 31T^{2} \)
37 \( 1 + (0.0508 + 0.0881i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.14 - 8.91i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.89 + 8.48i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.26T + 47T^{2} \)
53 \( 1 + (5.32 + 3.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.11T + 59T^{2} \)
61 \( 1 - 1.07iT - 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 12.4iT - 71T^{2} \)
73 \( 1 + (12.4 + 7.16i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + (4.32 - 7.49i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.05 - 8.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.78 - 5.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52568403733892064554493572148, −10.04922314325052913448491695989, −9.134813384884301928846266732427, −7.979714149494031265483449024431, −7.51290832427563729303124461834, −6.04021170648236084138432338252, −5.04540028330771016546668930392, −4.01077224767050769512016154818, −3.20551512343439383307884108067, −2.42694918678332936228241294718, 0.17784256476421120532762158345, 1.58880802044494469792554012635, 3.34577976885742959422518694893, 4.41420107754140134803511950920, 5.64108235712942109977572768755, 6.08397401946478818220519502646, 7.50615042626715835150230912836, 7.895298415170639791241376113618, 8.665379289243599922240228902612, 9.157393025306141336142389654019

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