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Properties

Label 2-129-43.35-c1-0-0
Degree $2$
Conductor $129$
Sign $-0.595 + 0.803i$
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  + (−0.612 + 2.68i)2-s + (−0.222 − 0.974i)3-s + (−5.03 − 2.42i)4-s + (−1.63 + 2.05i)5-s + 2.75·6-s − 2.70·7-s + (6.15 − 7.72i)8-s + (−0.900 + 0.433i)9-s + (−4.50 − 5.64i)10-s + (−1.76 + 0.851i)11-s + (−1.24 + 5.44i)12-s + (−0.728 + 0.913i)13-s + (1.65 − 7.26i)14-s + (2.36 + 1.13i)15-s + (9.99 + 12.5i)16-s + (3.95 + 4.96i)17-s + ⋯
L(s)  = 1  + (−0.433 + 1.89i)2-s + (−0.128 − 0.562i)3-s + (−2.51 − 1.21i)4-s + (−0.731 + 0.916i)5-s + 1.12·6-s − 1.02·7-s + (2.17 − 2.72i)8-s + (−0.300 + 0.144i)9-s + (−1.42 − 1.78i)10-s + (−0.533 + 0.256i)11-s + (−0.358 + 1.57i)12-s + (−0.202 + 0.253i)13-s + (0.443 − 1.94i)14-s + (0.609 + 0.293i)15-s + (2.49 + 3.13i)16-s + (0.959 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $-0.595 + 0.803i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ -0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.136380 - 0.271029i\)
\(L(\frac12)\) \(\approx\) \(0.136380 - 0.271029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (-5.70 + 3.24i)T \)
good2 \( 1 + (0.612 - 2.68i)T + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (1.63 - 2.05i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.70T + 7T^{2} \)
11 \( 1 + (1.76 - 0.851i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.728 - 0.913i)T + (-2.89 - 12.6i)T^{2} \)
17 \( 1 + (-3.95 - 4.96i)T + (-3.78 + 16.5i)T^{2} \)
19 \( 1 + (0.295 + 0.142i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (7.25 - 3.49i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (-0.646 + 2.83i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.351 + 1.54i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 1.56T + 37T^{2} \)
41 \( 1 + (0.656 - 2.87i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (2.07 + 0.998i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-8.36 - 10.4i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (1.99 + 2.49i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (1.55 + 6.82i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-8.03 - 3.86i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (8.21 + 3.95i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.74 - 2.18i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 5.62T + 79T^{2} \)
83 \( 1 + (-1.56 - 6.84i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-2.11 - 9.27i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-5.18 + 2.49i)T + (60.4 - 75.8i)T^{2} \)
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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

−14.27756486751220462565307582936, −13.35293293601328716917962049192, −12.27585788354009580731342366918, −10.47531785358071132242302110049, −9.596961308305734314520083793348, −8.110372545529328419508355945564, −7.49779458473794278930019059276, −6.53936885731162838118189989711, −5.72142229574340739578461710152, −3.85270750120787328856521623876, 0.37182324541509379201968918804, 2.92353782446688079429592978719, 4.03933423543047267605337827213, 5.20162308992709049349781663141, 7.911201334211505128693152303220, 8.874615025295302628670516742421, 9.817214979125357844072965731761, 10.48497318332901844389474345435, 11.78439441078329288273791062957, 12.29574479336784923948067579473

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