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Properties

Label 2-882-147.101-c1-0-1
Degree $2$
Conductor $882$
Sign $-0.819 - 0.573i$
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  + (−0.930 − 0.365i)2-s + (0.733 + 0.680i)4-s + (−0.668 − 0.455i)5-s + (1.56 + 2.13i)7-s + (−0.433 − 0.900i)8-s + (0.455 + 0.668i)10-s + (0.205 + 1.36i)11-s + (−4.63 + 3.70i)13-s + (−0.674 − 2.55i)14-s + (0.0747 + 0.997i)16-s + (−6.36 − 1.96i)17-s + (−0.313 + 0.180i)19-s + (−0.180 − 0.788i)20-s + (0.307 − 1.34i)22-s + (−0.919 − 2.98i)23-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (0.366 + 0.340i)4-s + (−0.298 − 0.203i)5-s + (0.590 + 0.806i)7-s + (−0.153 − 0.318i)8-s + (0.144 + 0.211i)10-s + (0.0620 + 0.411i)11-s + (−1.28 + 1.02i)13-s + (−0.180 − 0.683i)14-s + (0.0186 + 0.249i)16-s + (−1.54 − 0.476i)17-s + (−0.0718 + 0.0414i)19-s + (−0.0402 − 0.176i)20-s + (0.0655 − 0.287i)22-s + (−0.191 − 0.621i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0967428 + 0.306923i\)
\(L(\frac12)\) \(\approx\) \(0.0967428 + 0.306923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.930 + 0.365i)T \)
3 \( 1 \)
7 \( 1 + (-1.56 - 2.13i)T \)
good5 \( 1 + (0.668 + 0.455i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.205 - 1.36i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (4.63 - 3.70i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (6.36 + 1.96i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (0.313 - 0.180i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.919 + 2.98i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (-2.15 + 0.492i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (-4.11 - 2.37i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.607 - 0.564i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (10.1 - 4.86i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (9.83 + 4.73i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (2.32 - 5.91i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (-2.08 + 2.25i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-3.04 + 2.07i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (-4.15 - 4.47i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (0.0121 - 0.0210i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (14.0 + 3.20i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (5.89 - 2.31i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (-4.89 - 8.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.22 - 10.3i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-3.30 - 0.498i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 + 6.58iT - 97T^{2} \)
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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

−10.30489645239489842212121942029, −9.622233694528132413765036432114, −8.705206019228689480798700095281, −8.268666914446640204443401756913, −7.08608041145971862944424084765, −6.47796440412178908300747279186, −4.92954668636803480077100897538, −4.38932596378279624648927453569, −2.65810910935729378571248803603, −1.88069171239473712923526462930, 0.17902169217366514685501380834, 1.84267959441819213751999627004, 3.23679230777004862004958659190, 4.49688268155309811814467326105, 5.42099144486105968921400301078, 6.65041862154798610812519745357, 7.34050620357818976693075288543, 8.089648843464111373717104803368, 8.788117078937335083712224385608, 9.988992150967235901574709483880

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