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Properties

Label 2-129-129.62-c1-0-7
Degree $2$
Conductor $129$
Sign $0.881 + 0.472i$
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  + (−0.258 − 1.71i)3-s + (0.445 + 1.94i)4-s + (4.43 − 2.55i)7-s + (−2.86 + 0.884i)9-s + (3.22 − 1.26i)12-s + (−0.242 − 3.23i)13-s + (−3.60 + 1.73i)16-s + (−2.48 + 8.04i)19-s + (−5.52 − 6.92i)21-s + (−1.82 + 4.65i)25-s + (2.25 + 4.68i)27-s + (6.95 + 7.49i)28-s + (−0.879 − 2.24i)31-s + (−3 − 5.19i)36-s + (−8.28 − 4.78i)37-s + ⋯
L(s)  = 1  + (−0.149 − 0.988i)3-s + (0.222 + 0.974i)4-s + (1.67 − 0.966i)7-s + (−0.955 + 0.294i)9-s + (0.930 − 0.365i)12-s + (−0.0672 − 0.897i)13-s + (−0.900 + 0.433i)16-s + (−0.568 + 1.84i)19-s + (−1.20 − 1.51i)21-s + (−0.365 + 0.930i)25-s + (0.433 + 0.900i)27-s + (1.31 + 1.41i)28-s + (−0.157 − 0.402i)31-s + (−0.5 − 0.866i)36-s + (−1.36 − 0.786i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12486 - 0.282358i\)
\(L(\frac12)\) \(\approx\) \(1.12486 - 0.282358i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 1.71i)T \)
43 \( 1 + (6.55 - 0.112i)T \)
good2 \( 1 + (-0.445 - 1.94i)T^{2} \)
5 \( 1 + (1.82 - 4.65i)T^{2} \)
7 \( 1 + (-4.43 + 2.55i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (0.242 + 3.23i)T + (-12.8 + 1.93i)T^{2} \)
17 \( 1 + (-6.21 - 15.8i)T^{2} \)
19 \( 1 + (2.48 - 8.04i)T + (-15.6 - 10.7i)T^{2} \)
23 \( 1 + (-1.71 - 22.9i)T^{2} \)
29 \( 1 + (27.7 + 8.54i)T^{2} \)
31 \( 1 + (0.879 + 2.24i)T + (-22.7 + 21.0i)T^{2} \)
37 \( 1 + (8.28 + 4.78i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.12 + 39.9i)T^{2} \)
47 \( 1 + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (52.4 + 7.89i)T^{2} \)
59 \( 1 + (-36.7 + 46.1i)T^{2} \)
61 \( 1 + (-14.1 - 5.54i)T + (44.7 + 41.4i)T^{2} \)
67 \( 1 + (2.15 + 0.665i)T + (55.3 + 37.7i)T^{2} \)
71 \( 1 + (5.30 - 70.8i)T^{2} \)
73 \( 1 + (-4.23 + 0.317i)T + (72.1 - 10.8i)T^{2} \)
79 \( 1 + (-5.55 - 9.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-79.3 + 24.4i)T^{2} \)
89 \( 1 + (85.0 - 26.2i)T^{2} \)
97 \( 1 + (-1.46 + 6.43i)T + (-87.3 - 42.0i)T^{2} \)
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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

−13.16472901170038651847092117106, −12.23357186635285004586737240773, −11.38835289955029908045613927022, −10.52139174246923011176859695119, −8.381615219053947046166839254323, −7.895364589378192173785196457831, −7.06918269198119880268637563077, −5.42651836448152976696382492410, −3.78124946803131712648921463558, −1.80674298657420727160625500956, 2.20070493198065664713271425922, 4.63946925403758013433613844871, 5.22049094805593552428282074325, 6.58400946024421680496348568341, 8.484802465877643606194873175021, 9.203453462020323818963615954049, 10.45146956231294459674742965827, 11.33120480379141349580150631039, 11.83789940128132414002891707349, 13.85577905285381159426981636279

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