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Properties

Label 2-882-63.5-c1-0-33
Degree $2$
Conductor $882$
Sign $-0.999 - 0.0272i$
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 + (0.765 − 1.55i)3-s − 4-s + (0.474 + 0.821i)5-s + (−1.55 − 0.765i)6-s + i·8-s + (−1.82 − 2.37i)9-s + (0.821 − 0.474i)10-s + (−1.51 − 0.876i)11-s + (−0.765 + 1.55i)12-s + (−0.720 − 0.416i)13-s + (1.64 − 0.108i)15-s + 16-s + (−2.73 − 4.73i)17-s + (−2.37 + 1.82i)18-s + (−3.21 − 1.85i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.441 − 0.897i)3-s − 0.5·4-s + (0.212 + 0.367i)5-s + (−0.634 − 0.312i)6-s + 0.353i·8-s + (−0.609 − 0.792i)9-s + (0.259 − 0.150i)10-s + (−0.457 − 0.264i)11-s + (−0.220 + 0.448i)12-s + (−0.199 − 0.115i)13-s + (0.423 − 0.0280i)15-s + 0.250·16-s + (−0.663 − 1.14i)17-s + (−0.560 + 0.431i)18-s + (−0.736 − 0.425i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 - 0.0272i$
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: \((2,\ 882,\ (\ :1/2),\ -0.999 - 0.0272i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0163485 + 1.20029i\)
\(L(\frac12)\) \(\approx\) \(0.0163485 + 1.20029i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.765 + 1.55i)T \)
7 \( 1 \)
good5 \( 1 + (-0.474 - 0.821i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.51 + 0.876i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.720 + 0.416i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.73 + 4.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.21 + 1.85i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.888 + 0.513i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (6.58 - 3.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 11.0iT - 31T^{2} \)
37 \( 1 + (2.19 - 3.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.05 + 7.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.08 - 5.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.80T + 47T^{2} \)
53 \( 1 + (-1.88 + 1.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 1.09iT - 61T^{2} \)
67 \( 1 - 14.4T + 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-3.16 + 1.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 9.84T + 79T^{2} \)
83 \( 1 + (-2.41 - 4.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.15 - 3.74i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.90 + 2.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.577524607162664300430350511516, −9.009658102420229244438402343870, −8.053584254518023502359967186919, −7.23497815419087858330365590786, −6.35347623090485968425800215662, −5.29701990298973735440295274891, −4.03112843693724189404092456175, −2.75762234670720637365616274561, −2.21777837342060847800132971470, −0.51873710936308289069478258200, 2.03028889922859546147695971067, 3.51822771655704869385084073393, 4.43214456363098171549601816067, 5.24892241135211395228173434256, 6.10062160358422078875900247883, 7.27617610807951435200985229102, 8.173571699024960800299392217452, 8.892140002523407902712879882786, 9.474504020771050964816877825998, 10.50471085117061488270614681749

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