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Properties

Label 2-882-147.101-c1-0-14
Degree $2$
Conductor $882$
Sign $-0.191 + 0.981i$
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 + (0.432 + 0.294i)5-s + (−1.64 − 2.07i)7-s + (−0.433 − 0.900i)8-s + (−0.294 − 0.432i)10-s + (0.320 + 2.12i)11-s + (2.39 − 1.90i)13-s + (0.774 + 2.52i)14-s + (0.0747 + 0.997i)16-s + (−4.61 − 1.42i)17-s + (4.76 − 2.75i)19-s + (0.116 + 0.510i)20-s + (0.478 − 2.09i)22-s + (0.0351 + 0.113i)23-s + ⋯
L(s)  = 1  + (−0.658 − 0.258i)2-s + (0.366 + 0.340i)4-s + (0.193 + 0.131i)5-s + (−0.621 − 0.783i)7-s + (−0.153 − 0.318i)8-s + (−0.0932 − 0.136i)10-s + (0.0966 + 0.640i)11-s + (0.663 − 0.529i)13-s + (0.207 + 0.676i)14-s + (0.0186 + 0.249i)16-s + (−1.11 − 0.345i)17-s + (1.09 − 0.631i)19-s + (0.0260 + 0.114i)20-s + (0.101 − 0.446i)22-s + (0.00732 + 0.0237i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.191 + 0.981i$
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: \((2,\ 882,\ (\ :1/2),\ -0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.556045 - 0.674693i\)
\(L(\frac12)\) \(\approx\) \(0.556045 - 0.674693i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 2.07i)T \)
good5 \( 1 + (-0.432 - 0.294i)T + (1.82 + 4.65i)T^{2} \)
11 \( 1 + (-0.320 - 2.12i)T + (-10.5 + 3.24i)T^{2} \)
13 \( 1 + (-2.39 + 1.90i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + (4.61 + 1.42i)T + (14.0 + 9.57i)T^{2} \)
19 \( 1 + (-4.76 + 2.75i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.0351 - 0.113i)T + (-19.0 + 12.9i)T^{2} \)
29 \( 1 + (4.76 - 1.08i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.23 + 0.714i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.24 + 4.86i)T + (2.76 - 36.8i)T^{2} \)
41 \( 1 + (-0.904 + 0.435i)T + (25.5 - 32.0i)T^{2} \)
43 \( 1 + (3.78 + 1.82i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-4.33 + 11.0i)T + (-34.4 - 31.9i)T^{2} \)
53 \( 1 + (0.364 - 0.392i)T + (-3.96 - 52.8i)T^{2} \)
59 \( 1 + (-9.74 + 6.64i)T + (21.5 - 54.9i)T^{2} \)
61 \( 1 + (0.182 + 0.196i)T + (-4.55 + 60.8i)T^{2} \)
67 \( 1 + (-6.72 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.17 + 0.496i)T + (63.9 + 30.8i)T^{2} \)
73 \( 1 + (7.27 - 2.85i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (4.67 + 8.10i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.0574 - 0.0720i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (10.7 + 1.61i)T + (85.0 + 26.2i)T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−9.808291924339604477754330583642, −9.295744338844527745080133721545, −8.288551767642918685294824929064, −7.26617157039118736224643093608, −6.78041737667115930230259231251, −5.65142794700495295016089371947, −4.31424129618627453373492454889, −3.32947003235963439177675913143, −2.12405359473770611520553441038, −0.53864779353437908216081123584, 1.44311025600018624543387154390, 2.78308093751360672919097044656, 3.97340800394538938533272671712, 5.49406010659897809988437927903, 6.06528489839821393955471297937, 6.92627502163811946982999736790, 7.997049791256393512029862444252, 8.859879722837707575613965930795, 9.338371064724208886481885569198, 10.14014113917082744300051131353

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