Properties

Label 2-882-441.130-c1-0-15
Degree $2$
Conductor $882$
Sign $0.139 - 0.990i$
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  + (−0.826 + 0.563i)2-s + (1.57 + 0.715i)3-s + (0.365 − 0.930i)4-s + (−0.148 − 0.650i)5-s + (−1.70 + 0.297i)6-s + (2.62 + 0.362i)7-s + (0.222 + 0.974i)8-s + (1.97 + 2.25i)9-s + (0.489 + 0.454i)10-s + (−2.04 − 0.983i)11-s + (1.24 − 1.20i)12-s + (−5.34 + 3.64i)13-s + (−2.36 + 1.17i)14-s + (0.231 − 1.13i)15-s + (−0.733 − 0.680i)16-s + (1.98 + 5.05i)17-s + ⋯
L(s)  = 1  + (−0.584 + 0.398i)2-s + (0.910 + 0.413i)3-s + (0.182 − 0.465i)4-s + (−0.0664 − 0.291i)5-s + (−0.696 + 0.121i)6-s + (0.990 + 0.136i)7-s + (0.0786 + 0.344i)8-s + (0.658 + 0.752i)9-s + (0.154 + 0.143i)10-s + (−0.615 − 0.296i)11-s + (0.358 − 0.348i)12-s + (−1.48 + 1.00i)13-s + (−0.633 + 0.314i)14-s + (0.0597 − 0.292i)15-s + (−0.183 − 0.170i)16-s + (0.480 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24130 + 1.07899i\)
\(L(\frac12)\) \(\approx\) \(1.24130 + 1.07899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 - 0.563i)T \)
3 \( 1 + (-1.57 - 0.715i)T \)
7 \( 1 + (-2.62 - 0.362i)T \)
good5 \( 1 + (0.148 + 0.650i)T + (-4.50 + 2.16i)T^{2} \)
11 \( 1 + (2.04 + 0.983i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (5.34 - 3.64i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (-1.98 - 5.05i)T + (-12.4 + 11.5i)T^{2} \)
19 \( 1 + (-3.77 - 6.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.24 + 6.57i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.30 - 0.498i)T + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (-2.25 - 3.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.29 - 0.194i)T + (35.3 + 10.9i)T^{2} \)
41 \( 1 + (-6.92 - 2.13i)T + (33.8 + 23.0i)T^{2} \)
43 \( 1 + (-3.50 + 1.08i)T + (35.5 - 24.2i)T^{2} \)
47 \( 1 + (-7.65 + 5.21i)T + (17.1 - 43.7i)T^{2} \)
53 \( 1 + (5.18 - 0.781i)T + (50.6 - 15.6i)T^{2} \)
59 \( 1 + (7.18 - 2.21i)T + (48.7 - 33.2i)T^{2} \)
61 \( 1 + (0.844 + 2.15i)T + (-44.7 + 41.4i)T^{2} \)
67 \( 1 + (-1.47 - 2.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.29 + 9.14i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.555 + 7.41i)T + (-72.1 - 10.8i)T^{2} \)
79 \( 1 + (0.780 - 1.35i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.747 + 0.509i)T + (30.3 + 77.2i)T^{2} \)
89 \( 1 + (4.63 + 3.15i)T + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (6.54 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.27698848135304942938949480638, −9.374130379198244591400483175218, −8.459255616941090433494898098486, −8.031255122622026662406881441109, −7.33591566508990697887239973854, −6.00778825227481060303308378358, −4.90514151668607605531115556361, −4.21031347879916463182862416815, −2.63854353079114732667795767432, −1.60930638367762949399502893316, 0.917748755042888742833799342632, 2.49123315753624205433547988871, 2.92127034094617520406327274012, 4.47231051252857399244108092925, 5.42174853546864902896683193795, 7.21379743686547015167049977241, 7.50481691430786500631290641070, 8.056547029778910596085450239301, 9.320308467107111753074212181650, 9.699834678869265670883875087822

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