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Properties

Label 2-882-63.5-c1-0-32
Degree $2$
Conductor $882$
Sign $-0.832 + 0.554i$
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 + (1.71 − 0.211i)3-s − 4-s + (−0.584 − 1.01i)5-s + (−0.211 − 1.71i)6-s + i·8-s + (2.91 − 0.725i)9-s + (−1.01 + 0.584i)10-s + (−4.99 − 2.88i)11-s + (−1.71 + 0.211i)12-s + (0.571 + 0.329i)13-s + (−1.21 − 1.61i)15-s + 16-s + (−2.83 − 4.91i)17-s + (−0.725 − 2.91i)18-s + (−3.16 − 1.82i)19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.992 − 0.121i)3-s − 0.5·4-s + (−0.261 − 0.453i)5-s + (−0.0861 − 0.701i)6-s + 0.353i·8-s + (0.970 − 0.241i)9-s + (−0.320 + 0.184i)10-s + (−1.50 − 0.869i)11-s + (−0.496 + 0.0609i)12-s + (0.158 + 0.0914i)13-s + (−0.314 − 0.417i)15-s + 0.250·16-s + (−0.688 − 1.19i)17-s + (−0.171 − 0.686i)18-s + (−0.726 − 0.419i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.832 + 0.554i$
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: \((2,\ 882,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.452479 - 1.49621i\)
\(L(\frac12)\) \(\approx\) \(0.452479 - 1.49621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.211i)T \)
7 \( 1 \)
good5 \( 1 + (0.584 + 1.01i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.99 + 2.88i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.571 - 0.329i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.83 + 4.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.16 + 1.82i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.503 - 0.290i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.53 + 3.77i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 + (-5.29 + 9.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.64 - 4.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.37 - 4.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.34T + 47T^{2} \)
53 \( 1 + (4.14 - 2.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.02T + 59T^{2} \)
61 \( 1 - 11.7iT - 61T^{2} \)
67 \( 1 - 8.52T + 67T^{2} \)
71 \( 1 + 8.34iT - 71T^{2} \)
73 \( 1 + (-0.899 + 0.519i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.26T + 79T^{2} \)
83 \( 1 + (-6.21 - 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.83 + 13.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (12.5 - 7.26i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.762350498809638423328092845740, −8.875963460255720516489470678674, −8.322268245628118503495201506717, −7.60528827870727583102044490215, −6.38698199088454182903753079824, −4.99099034787683449819137791911, −4.28776150721714179140323766808, −2.98255415134314223038923404007, −2.39251230793244034242213288054, −0.65067087940410048583532796430, 2.02052810950124043849689179004, 3.16469707828911942228551001769, 4.25267478843601283049839663081, 5.08823688471396120007471340741, 6.43515929011849488915096624961, 7.16073358684866646466109515783, 8.168316138573609483057291200172, 8.349098264877179823284024039176, 9.601800034767665976862694609967, 10.31869662550207585640981047326

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