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Properties

Label 2-129-129.98-c1-0-2
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: $\chi_{129} (98, \cdot )$
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

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

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