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Properties

Label 2-129-129.65-c1-0-9
Degree $2$
Conductor $129$
Sign $-0.678 + 0.734i$
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  + (−0.754 − 0.363i)2-s + (0.0167 − 1.73i)3-s + (−0.810 − 1.01i)4-s + (0.252 + 1.10i)5-s + (−0.641 + 1.30i)6-s − 2.80i·7-s + (0.614 + 2.69i)8-s + (−2.99 − 0.0579i)9-s + (0.211 − 0.927i)10-s + (−4.14 − 3.30i)11-s + (−1.77 + 1.38i)12-s + (0.0697 + 0.305i)13-s + (−1.01 + 2.11i)14-s + (1.92 − 0.419i)15-s + (−0.0638 + 0.279i)16-s + (6.26 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.533 − 0.256i)2-s + (0.00965 − 0.999i)3-s + (−0.405 − 0.507i)4-s + (0.113 + 0.495i)5-s + (−0.261 + 0.530i)6-s − 1.06i·7-s + (0.217 + 0.951i)8-s + (−0.999 − 0.0193i)9-s + (0.0669 − 0.293i)10-s + (−1.24 − 0.995i)11-s + (−0.511 + 0.400i)12-s + (0.0193 + 0.0847i)13-s + (−0.272 + 0.565i)14-s + (0.496 − 0.108i)15-s + (−0.0159 + 0.0699i)16-s + (1.51 + 0.346i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.678 + 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.678 + 0.734i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ -0.678 + 0.734i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270025 - 0.617049i\)
\(L(\frac12)\) \(\approx\) \(0.270025 - 0.617049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0167 + 1.73i)T \)
43 \( 1 + (6.12 + 2.34i)T \)
good2 \( 1 + (0.754 + 0.363i)T + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (-0.252 - 1.10i)T + (-4.50 + 2.16i)T^{2} \)
7 \( 1 + 2.80iT - 7T^{2} \)
11 \( 1 + (4.14 + 3.30i)T + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-0.0697 - 0.305i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (-6.26 - 1.43i)T + (15.3 + 7.37i)T^{2} \)
19 \( 1 + (-3.39 + 2.70i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (4.35 + 3.47i)T + (5.11 + 22.4i)T^{2} \)
29 \( 1 + (-3.75 - 1.80i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-8.10 - 3.90i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 - 1.67iT - 37T^{2} \)
41 \( 1 + (-1.16 + 2.42i)T + (-25.5 - 32.0i)T^{2} \)
47 \( 1 + (2.76 - 2.20i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (6.95 + 1.58i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 + (-6.52 - 1.48i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (0.640 + 1.33i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-2.64 - 3.32i)T + (-14.9 + 65.3i)T^{2} \)
71 \( 1 + (2.17 + 2.72i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (7.52 - 1.71i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + 0.979T + 79T^{2} \)
83 \( 1 + (-0.764 - 1.58i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (-0.326 + 0.157i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-2.01 + 2.53i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.15887845079642171740665702114, −11.80663087736174890425795529208, −10.61727730333032768046849938643, −10.16850174048284224430900247636, −8.481341515103344806975887815602, −7.76450267691106175470403756626, −6.43787817866257319548067026539, −5.19267140096866033299116523754, −2.94091182251793338789793392107, −0.898935398806902026132106478775, 3.06398657045693764920372521546, 4.71554800360893592569407756547, 5.64653280733519237850949160216, 7.74481146273415305817792642931, 8.437794921253849169950444372384, 9.783316872654048803017794937051, 9.904650368236088539201905563302, 11.82150709007034612453158929432, 12.50721096410418090817147932990, 13.72058996845041051660469018464

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