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Properties

Label 2-129-129.32-c1-0-2
Degree $2$
Conductor $129$
Sign $-0.0939 + 0.995i$
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  + (−1.54 − 1.93i)2-s + (−0.980 + 1.42i)3-s + (−0.919 + 4.02i)4-s + (2.61 − 1.26i)5-s + (4.27 − 0.305i)6-s − 2.05i·7-s + (4.75 − 2.29i)8-s + (−1.07 − 2.80i)9-s + (−6.47 − 3.12i)10-s + (−0.179 + 0.0409i)11-s + (−4.84 − 5.26i)12-s + (5.16 − 2.48i)13-s + (−3.98 + 3.18i)14-s + (−0.767 + 4.97i)15-s + (−4.33 − 2.08i)16-s + (0.844 − 1.75i)17-s + ⋯
L(s)  = 1  + (−1.09 − 1.36i)2-s + (−0.566 + 0.824i)3-s + (−0.459 + 2.01i)4-s + (1.17 − 0.563i)5-s + (1.74 − 0.124i)6-s − 0.778i·7-s + (1.68 − 0.809i)8-s + (−0.358 − 0.933i)9-s + (−2.04 − 0.986i)10-s + (−0.0541 + 0.0123i)11-s + (−1.39 − 1.51i)12-s + (1.43 − 0.690i)13-s + (−1.06 + 0.850i)14-s + (−0.198 + 1.28i)15-s + (−1.08 − 0.521i)16-s + (0.204 − 0.425i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0939 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0939 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.0939 + 0.995i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ -0.0939 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416734 - 0.457927i\)
\(L(\frac12)\) \(\approx\) \(0.416734 - 0.457927i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.980 - 1.42i)T \)
43 \( 1 + (6.06 - 2.49i)T \)
good2 \( 1 + (1.54 + 1.93i)T + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-2.61 + 1.26i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + 2.05iT - 7T^{2} \)
11 \( 1 + (0.179 - 0.0409i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-5.16 + 2.48i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + (-0.844 + 1.75i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + (6.46 + 1.47i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-6.67 + 1.52i)T + (20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.595 + 0.746i)T + (-6.45 + 28.2i)T^{2} \)
31 \( 1 + (-3.43 - 4.30i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 - 7.76iT - 37T^{2} \)
41 \( 1 + (5.19 - 4.14i)T + (9.12 - 39.9i)T^{2} \)
47 \( 1 + (4.76 + 1.08i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.227 - 0.472i)T + (-33.0 - 41.4i)T^{2} \)
59 \( 1 + (5.76 - 11.9i)T + (-36.7 - 46.1i)T^{2} \)
61 \( 1 + (-1.20 - 0.961i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.34 + 5.89i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-0.756 + 3.31i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.40 + 7.08i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 + (1.33 + 1.06i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.01 + 2.53i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-0.220 - 0.967i)T + (-87.3 + 42.0i)T^{2} \)
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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

−12.90961415051373968162599705884, −11.60414646156073209526876461082, −10.59718652207547408625941794589, −10.26710152614671393770355247699, −9.159950830021168777233627849132, −8.452295628349267187946391995443, −6.37097056921805245113453626509, −4.76707393479788038515260830649, −3.23058822977022025826272408364, −1.14247675286093439147978100910, 1.82919266981180636581386207396, 5.50213408405408683751506171844, 6.23476364724304952776770170034, 6.82041673870463354491078633480, 8.283138374754043833671775791907, 9.042174570550752493714010068223, 10.30778826341027414618128002850, 11.20733094914000992436149026539, 12.85089755134728707122479666210, 13.81443934679708469405786010024

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