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Properties

Label 2-129-129.8-c1-0-11
Degree $2$
Conductor $129$
Sign $-0.997 - 0.0745i$
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  + (0.554 − 2.42i)2-s + (−1.66 − 0.481i)3-s + (−3.78 − 1.82i)4-s + (0.610 − 0.765i)5-s + (−2.08 + 3.77i)6-s − 0.100i·7-s + (−3.41 + 4.28i)8-s + (2.53 + 1.60i)9-s + (−1.52 − 1.90i)10-s + (−1.23 − 2.57i)11-s + (5.42 + 4.85i)12-s + (−2.33 + 2.92i)13-s + (−0.243 − 0.0556i)14-s + (−1.38 + 0.980i)15-s + (3.26 + 4.09i)16-s + (6.00 − 4.79i)17-s + ⋯
L(s)  = 1  + (0.391 − 1.71i)2-s + (−0.960 − 0.277i)3-s + (−1.89 − 0.911i)4-s + (0.273 − 0.342i)5-s + (−0.853 + 1.54i)6-s − 0.0379i·7-s + (−1.20 + 1.51i)8-s + (0.845 + 0.533i)9-s + (−0.480 − 0.602i)10-s + (−0.373 − 0.775i)11-s + (1.56 + 1.40i)12-s + (−0.647 + 0.812i)13-s + (−0.0651 − 0.0148i)14-s + (−0.357 + 0.253i)15-s + (0.817 + 1.02i)16-s + (1.45 − 1.16i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.997 - 0.0745i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ -0.997 - 0.0745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0331915 + 0.889093i\)
\(L(\frac12)\) \(\approx\) \(0.0331915 + 0.889093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.66 + 0.481i)T \)
43 \( 1 + (-6.55 - 0.293i)T \)
good2 \( 1 + (-0.554 + 2.42i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (-0.610 + 0.765i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 0.100iT - 7T^{2} \)
11 \( 1 + (1.23 + 2.57i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.33 - 2.92i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-6.00 + 4.79i)T + (3.78 - 16.5i)T^{2} \)
19 \( 1 + (-3.16 + 6.56i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.598 - 1.24i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (-0.720 + 3.15i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.22 - 5.35i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 7.93iT - 37T^{2} \)
41 \( 1 + (-3.10 - 0.709i)T + (36.9 + 17.7i)T^{2} \)
47 \( 1 + (0.571 - 1.18i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (7.77 - 6.19i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 + (-0.284 + 0.226i)T + (13.1 - 57.5i)T^{2} \)
61 \( 1 + (-9.27 + 2.11i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (0.580 + 0.279i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-0.471 - 0.227i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (5.49 + 4.38i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + 4.42T + 79T^{2} \)
83 \( 1 + (7.11 - 1.62i)T + (74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.11 + 4.87i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (8.79 - 4.23i)T + (60.4 - 75.8i)T^{2} \)
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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

−12.52780767269654781206310576397, −11.69413499809048620902225258559, −11.14773022962551288793644778377, −9.978519630726990166887843628605, −9.209066576558415187376690671702, −7.28043942728258371355075347342, −5.47091394222006487098707097922, −4.72011745070209984470280602616, −2.90704018874113039578329099025, −1.04260057868314271390076469320, 3.99344529597815706285881633554, 5.41186201713058608798113680133, 5.90682897309722240468465618853, 7.22243393453553692248564068940, 7.997699006885117480848733135345, 9.729303622325503817807128217849, 10.47578258901562551497157811492, 12.33938225229506025638280443510, 12.77882641848507390475054023304, 14.40963716934680123510390190224

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