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Properties

Label 2-129-129.113-c1-0-8
Degree $2$
Conductor $129$
Sign $0.470 + 0.882i$
Analytic cond. $1.03007$
Root an. cond. $1.01492$
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  + (0.178 + 0.783i)2-s + (−1.60 − 0.657i)3-s + (1.22 − 0.587i)4-s + (−2.53 − 3.17i)5-s + (0.228 − 1.37i)6-s − 3.17i·7-s + (1.68 + 2.10i)8-s + (2.13 + 2.10i)9-s + (2.03 − 2.55i)10-s + (−0.516 + 1.07i)11-s + (−2.34 + 0.139i)12-s + (0.593 + 0.744i)13-s + (2.48 − 0.567i)14-s + (1.96 + 6.75i)15-s + (0.338 − 0.425i)16-s + (0.109 + 0.0872i)17-s + ⋯
L(s)  = 1  + (0.126 + 0.553i)2-s + (−0.925 − 0.379i)3-s + (0.610 − 0.293i)4-s + (−1.13 − 1.41i)5-s + (0.0933 − 0.560i)6-s − 1.20i·7-s + (0.594 + 0.744i)8-s + (0.711 + 0.702i)9-s + (0.643 − 0.806i)10-s + (−0.155 + 0.323i)11-s + (−0.676 + 0.0401i)12-s + (0.164 + 0.206i)13-s + (0.664 − 0.151i)14-s + (0.508 + 1.74i)15-s + (0.0847 − 0.106i)16-s + (0.0265 + 0.0211i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.734883 - 0.441006i\)
\(L(\frac12)\) \(\approx\) \(0.734883 - 0.441006i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.60 + 0.657i)T \)
43 \( 1 + (5.93 + 2.79i)T \)
good2 \( 1 + (-0.178 - 0.783i)T + (-1.80 + 0.867i)T^{2} \)
5 \( 1 + (2.53 + 3.17i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + 3.17iT - 7T^{2} \)
11 \( 1 + (0.516 - 1.07i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-0.593 - 0.744i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + (-0.109 - 0.0872i)T + (3.78 + 16.5i)T^{2} \)
19 \( 1 + (0.157 + 0.326i)T + (-11.8 + 14.8i)T^{2} \)
23 \( 1 + (-3.80 + 7.89i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-1.25 - 5.50i)T + (-26.1 + 12.5i)T^{2} \)
31 \( 1 + (-1.51 - 6.62i)T + (-27.9 + 13.4i)T^{2} \)
37 \( 1 + 1.16iT - 37T^{2} \)
41 \( 1 + (-6.60 + 1.50i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-0.952 - 1.97i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (-6.04 - 4.82i)T + (11.7 + 51.6i)T^{2} \)
59 \( 1 + (5.49 + 4.37i)T + (13.1 + 57.5i)T^{2} \)
61 \( 1 + (-5.92 - 1.35i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-4.43 + 2.13i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-8.40 + 4.04i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (5.34 - 4.26i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 6.11T + 79T^{2} \)
83 \( 1 + (12.3 + 2.82i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.425 - 1.86i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-2.31 - 1.11i)T + (60.4 + 75.8i)T^{2} \)
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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

−12.90139138507422096583811198345, −12.22670397725395687579438642297, −11.15958465235524839065529498083, −10.42540331484782855051984724775, −8.539978302929549691478207975173, −7.45945918251168676587158512380, −6.75486951018723485441935304064, −5.15726559467356303919657401108, −4.37761776226428949213344174721, −1.05579041382273639545417553940, 2.80686565075073457549393000945, 3.91743659099466713009268857257, 5.81201003179310955251627135321, 6.88457305012781058000998163102, 7.937019103754489776752012710879, 9.784636340845398478377587981939, 10.86550475064617492192726773721, 11.57923975419009666356729692183, 11.84962583688664077749471024989, 13.12866758305965158185182314936

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