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Properties

Label 2-129-129.104-c1-0-10
Degree $2$
Conductor $129$
Sign $0.288 + 0.957i$
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  + (0.975 − 1.43i)3-s + (−1.24 − 1.56i)4-s + (0.157 + 0.0906i)7-s + (−1.09 − 2.79i)9-s + (−3.45 + 0.258i)12-s + (2.87 + 0.887i)13-s + (−0.890 + 3.89i)16-s + (6.13 + 2.40i)19-s + (0.282 − 0.136i)21-s + (−0.373 + 4.98i)25-s + (−5.06 − 1.15i)27-s + (−0.0540 − 0.358i)28-s + (−0.190 − 2.54i)31-s + (−3 + 5.19i)36-s + (−10.2 + 5.91i)37-s + ⋯
L(s)  = 1  + (0.563 − 0.826i)3-s + (−0.623 − 0.781i)4-s + (0.0593 + 0.0342i)7-s + (−0.365 − 0.930i)9-s + (−0.997 + 0.0747i)12-s + (0.797 + 0.246i)13-s + (−0.222 + 0.974i)16-s + (1.40 + 0.552i)19-s + (0.0617 − 0.0297i)21-s + (−0.0747 + 0.997i)25-s + (−0.974 − 0.222i)27-s + (−0.0102 − 0.0677i)28-s + (−0.0342 − 0.456i)31-s + (−0.5 + 0.866i)36-s + (−1.68 + 0.973i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.288 + 0.957i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.288 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.905589 - 0.672877i\)
\(L(\frac12)\) \(\approx\) \(0.905589 - 0.672877i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.975 + 1.43i)T \)
43 \( 1 + (-5.85 - 2.94i)T \)
good2 \( 1 + (1.24 + 1.56i)T^{2} \)
5 \( 1 + (0.373 - 4.98i)T^{2} \)
7 \( 1 + (-0.157 - 0.0906i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.87 - 0.887i)T + (10.7 + 7.32i)T^{2} \)
17 \( 1 + (-1.27 - 16.9i)T^{2} \)
19 \( 1 + (-6.13 - 2.40i)T + (13.9 + 12.9i)T^{2} \)
23 \( 1 + (-21.9 - 6.77i)T^{2} \)
29 \( 1 + (10.5 - 26.9i)T^{2} \)
31 \( 1 + (0.190 + 2.54i)T + (-30.6 + 4.62i)T^{2} \)
37 \( 1 + (10.2 - 5.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-25.5 - 32.0i)T^{2} \)
47 \( 1 + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-43.7 + 29.8i)T^{2} \)
59 \( 1 + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (6.60 + 0.495i)T + (60.3 + 9.09i)T^{2} \)
67 \( 1 + (-5.95 + 15.1i)T + (-49.1 - 45.5i)T^{2} \)
71 \( 1 + (67.8 - 20.9i)T^{2} \)
73 \( 1 + (-3.24 + 10.5i)T + (-60.3 - 41.1i)T^{2} \)
79 \( 1 + (7.99 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-30.3 - 77.2i)T^{2} \)
89 \( 1 + (32.5 + 82.8i)T^{2} \)
97 \( 1 + (10.3 - 12.9i)T + (-21.5 - 94.5i)T^{2} \)
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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

−13.46354241243896724618475218729, −12.25384957283805433696648189648, −11.12318764607720398530037874739, −9.755966894860328950997522551075, −8.924267900864037407919355807191, −7.83516928978104491049877104662, −6.50770491669707357231514022967, −5.34979997902590479045994575977, −3.56141171491577620501637890913, −1.45732092436079162203004554877, 3.03100356491291245024452452250, 4.13853602308160976916171988263, 5.38000288169988713017573130684, 7.37002560419912259131914658074, 8.453039913286992529339815951879, 9.186744282382360553702023856813, 10.28960639216258166893715976389, 11.45164294951691541381807318157, 12.64333356955762173906902639640, 13.76819463734439575547679250590

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