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Properties

Label 2-882-441.151-c1-0-55
Degree $2$
Conductor $882$
Sign $-0.712 - 0.701i$
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  + (0.900 − 0.433i)2-s + (0.0687 − 1.73i)3-s + (0.623 − 0.781i)4-s + (−2.95 − 2.74i)5-s + (−0.689 − 1.58i)6-s + (−2.62 − 0.309i)7-s + (0.222 − 0.974i)8-s + (−2.99 − 0.237i)9-s + (−3.85 − 1.18i)10-s + (−1.43 − 0.975i)11-s + (−1.31 − 1.13i)12-s + (5.52 + 3.76i)13-s + (−2.50 + 0.860i)14-s + (−4.95 + 4.93i)15-s + (−0.222 − 0.974i)16-s + (6.45 − 0.972i)17-s + ⋯
L(s)  = 1  + (0.637 − 0.306i)2-s + (0.0396 − 0.999i)3-s + (0.311 − 0.390i)4-s + (−1.32 − 1.22i)5-s + (−0.281 − 0.648i)6-s + (−0.993 − 0.117i)7-s + (0.0786 − 0.344i)8-s + (−0.996 − 0.0792i)9-s + (−1.21 − 0.376i)10-s + (−0.431 − 0.294i)11-s + (−0.378 − 0.327i)12-s + (1.53 + 1.04i)13-s + (−0.668 + 0.230i)14-s + (−1.27 + 1.27i)15-s + (−0.0556 − 0.243i)16-s + (1.56 − 0.235i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ -0.712 - 0.701i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.362470 + 0.885241i\)
\(L(\frac12)\) \(\approx\) \(0.362470 + 0.885241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 + 0.433i)T \)
3 \( 1 + (-0.0687 + 1.73i)T \)
7 \( 1 + (2.62 + 0.309i)T \)
good5 \( 1 + (2.95 + 2.74i)T + (0.373 + 4.98i)T^{2} \)
11 \( 1 + (1.43 + 0.975i)T + (4.01 + 10.2i)T^{2} \)
13 \( 1 + (-5.52 - 3.76i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (-6.45 + 0.972i)T + (16.2 - 5.01i)T^{2} \)
19 \( 1 + (0.243 + 0.422i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.06 + 2.70i)T + (-16.8 + 15.6i)T^{2} \)
29 \( 1 + (8.40 - 1.26i)T + (27.7 - 8.54i)T^{2} \)
31 \( 1 + 9.43T + 31T^{2} \)
37 \( 1 + (-1.47 + 3.74i)T + (-27.1 - 25.1i)T^{2} \)
41 \( 1 + (3.49 - 1.07i)T + (33.8 - 23.0i)T^{2} \)
43 \( 1 + (1.14 + 0.351i)T + (35.5 + 24.2i)T^{2} \)
47 \( 1 + (-6.68 + 3.22i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (4.02 + 10.2i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (1.04 + 4.58i)T + (-53.1 + 25.5i)T^{2} \)
61 \( 1 + (3.88 + 4.87i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + 8.16T + 67T^{2} \)
71 \( 1 + (-1.00 + 1.26i)T + (-15.7 - 69.2i)T^{2} \)
73 \( 1 + (-4.03 + 2.75i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + 6.03T + 79T^{2} \)
83 \( 1 + (-0.582 + 0.396i)T + (30.3 - 77.2i)T^{2} \)
89 \( 1 + (-0.826 - 11.0i)T + (-88.0 + 13.2i)T^{2} \)
97 \( 1 + (-1.64 + 2.85i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.371836508639605548470830459034, −8.722896159102057334802993933847, −7.82052821543031164641719176050, −7.09462245568427124571509106014, −6.01462003722460387208749903223, −5.25881750064448989324443265970, −3.81990915875965747984014833509, −3.42590168278898724502947856764, −1.61635292240272355470252294107, −0.37257578819027879204027701817, 3.01156440933678154262753324885, 3.46709157331382234281934014835, 4.02059389140075650321148244442, 5.60906260042313086992624240899, 6.05073664810855636750135708720, 7.41002855519191770830937398913, 7.83922336425819948304677227240, 8.941580405781059284420692426391, 10.12750207332161393650126644361, 10.67729681255029218079292475626

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