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Properties

Label 2-882-63.5-c1-0-14
Degree $2$
Conductor $882$
Sign $0.317 + 0.948i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.317 + 0.948i$
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: \((2,\ 882,\ (\ :1/2),\ 0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.782761 - 0.563395i\)
\(L(\frac12)\) \(\approx\) \(0.782761 - 0.563395i\)
\(L(\frac{3}{2})\) 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

−10.30347815084963651208811390050, −9.545611582770298755526927338408, −8.213656012499755109840243266809, −7.54612240015156914105874778008, −6.09476179493271040627779100682, −5.70054883081947301641520447137, −4.62205071224589119809131316030, −3.48240924210925707291895725275, −2.34221156109408054384107132822, −0.68897127851867564487835918316, 1.02490009667103144809392885538, 2.82425755754221143942843006396, 4.70297795520873097637095860665, 5.02368027034372281120001989887, 5.81305622058764082768384080054, 7.04799864382237204767866329295, 7.42017478980370012983722946328, 8.476918684139690279891879292925, 9.701264127857342672433247027809, 10.05079182503123963876709467651

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