Properties

Label 2-882-21.2-c2-0-20
Degree $2$
Conductor $882$
Sign $-0.230 + 0.973i$
Analytic cond. $24.0327$
Root an. cond. $4.90232$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (1.73 + i)5-s − 2.82i·8-s + (−1.41 − 2.44i)10-s + (−2.44 + 1.41i)11-s − 12.7·13-s + (−2.00 + 3.46i)16-s + (19.0 − 11i)17-s + (−2.82 + 4.89i)19-s + 3.99i·20-s + 4·22-s + (−2.44 − 1.41i)23-s + (−10.5 − 18.1i)25-s + (15.5 + 9i)26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.346 + 0.200i)5-s − 0.353i·8-s + (−0.141 − 0.244i)10-s + (−0.222 + 0.128i)11-s − 0.979·13-s + (−0.125 + 0.216i)16-s + (1.12 − 0.647i)17-s + (−0.148 + 0.257i)19-s + 0.199i·20-s + 0.181·22-s + (−0.106 − 0.0614i)23-s + (−0.419 − 0.727i)25-s + (0.599 + 0.346i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.230 + 0.973i$
Analytic conductor: \(24.0327\)
Root analytic conductor: \(4.90232\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1),\ -0.230 + 0.973i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9812427981\)
\(L(\frac12)\) \(\approx\) \(0.9812427981\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.22 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.73 - i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.44 - 1.41i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + (-19.0 + 11i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (2.82 - 4.89i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (2.44 + 1.41i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 35.3iT - 841T^{2} \)
31 \( 1 + (16.9 + 29.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (32 - 55.4i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 20iT - 1.68e3T^{2} \)
43 \( 1 - 44T + 1.84e3T^{2} \)
47 \( 1 + (-58.8 - 34i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-15.9 + 9.19i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-86.6 + 50i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-26.1 + 45.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (60 + 103. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 8.48iT - 5.04e3T^{2} \)
73 \( 1 + (37.4 + 64.9i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (46 - 79.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 112iT - 6.88e3T^{2} \)
89 \( 1 + (17.3 + 10i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 26.8T + 9.40e3T^{2} \)
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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

−9.972096096963701398991699799216, −9.030145076754917773446743828339, −7.922280535772723936098275209158, −7.42380519852194579890106405065, −6.31086396715613728089263645429, −5.34560010183093785620454324769, −4.16737636708386914206203239665, −2.89941521627998850680325069264, −2.00399559504466543748986418724, −0.41861073015580665067746852900, 1.21609628755011426779884943061, 2.46834153037329884873032179840, 3.82439344145330662292456878847, 5.28304554722596141991797654026, 5.69252373048059106961687727802, 7.05878830021578517709544232783, 7.51255973347176535256696648015, 8.652050151346061781491927885709, 9.204696630173400048671196569309, 10.21060075665683782078895705205

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