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Properties

Label 2-882-63.5-c1-0-34
Degree $2$
Conductor $882$
Sign $-0.0136 + 0.999i$
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 + (1.63 − 0.568i)3-s − 4-s + (−1.55 − 2.69i)5-s + (0.568 + 1.63i)6-s − i·8-s + (2.35 − 1.86i)9-s + (2.69 − 1.55i)10-s + (1.07 + 0.619i)11-s + (−1.63 + 0.568i)12-s + (−4.97 − 2.87i)13-s + (−4.07 − 3.51i)15-s + 16-s + (−0.783 − 1.35i)17-s + (1.86 + 2.35i)18-s + (−5.82 − 3.36i)19-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.944 − 0.328i)3-s − 0.5·4-s + (−0.694 − 1.20i)5-s + (0.232 + 0.667i)6-s − 0.353i·8-s + (0.784 − 0.620i)9-s + (0.851 − 0.491i)10-s + (0.323 + 0.186i)11-s + (−0.472 + 0.164i)12-s + (−1.37 − 0.796i)13-s + (−1.05 − 0.908i)15-s + 0.250·16-s + (−0.190 − 0.329i)17-s + (0.438 + 0.554i)18-s + (−1.33 − 0.771i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0136 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0136 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0136 + 0.999i$
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: \((2,\ 882,\ (\ :1/2),\ -0.0136 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900484 - 0.912843i\)
\(L(\frac12)\) \(\approx\) \(0.900484 - 0.912843i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 0.568i)T \)
7 \( 1 \)
good5 \( 1 + (1.55 + 2.69i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.07 - 0.619i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.97 + 2.87i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.783 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.82 + 3.36i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.52 - 0.879i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.46 - 2.00i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.94iT - 31T^{2} \)
37 \( 1 + (-2.63 + 4.55i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.70 + 9.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.38 + 2.40i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.263T + 47T^{2} \)
53 \( 1 + (-10.4 + 6.05i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 9.36iT - 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 - 0.534iT - 71T^{2} \)
73 \( 1 + (-8.05 + 4.65i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + (-4.82 - 8.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.06 - 5.30i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.29 - 0.750i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.507898799667420938903779915744, −8.815184580611690950154278008552, −8.325384906058330287705221598414, −7.39532329613261971334997091619, −6.88531515255781150187008150430, −5.37933583985999502626758976904, −4.58519399971369842549505516605, −3.72256620264960899782255294377, −2.25319128911859958102157992245, −0.52932455177774490517146712478, 2.10533746018169817744743167493, 2.80221855301407427781370955366, 4.01948446027082626888977369586, 4.35046069279598029855500449506, 6.11785663408953388702485308416, 7.20096073739801290092339172305, 7.88127260553650379716731328757, 8.747136226051345425208875114552, 9.783149208566944235347143817918, 10.17540922072101872273682483619

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