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Properties

Label 2-129-129.116-c1-0-6
Degree $2$
Conductor $129$
Sign $0.790 - 0.613i$
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  + (1.17 + 1.26i)3-s + (1.80 + 0.867i)4-s + (−3.48 − 2.01i)7-s + (−0.224 + 2.99i)9-s + (1.02 + 3.31i)12-s + (−2.02 − 5.17i)13-s + (2.49 + 3.12i)16-s + (7.00 − 0.525i)19-s + (−1.54 − 6.79i)21-s + (−4.77 − 1.47i)25-s + (−4.06 + 3.23i)27-s + (−4.53 − 6.64i)28-s + (−9.55 + 2.94i)31-s + (−3 + 5.19i)36-s + (8.18 − 4.72i)37-s + ⋯
L(s)  = 1  + (0.680 + 0.733i)3-s + (0.900 + 0.433i)4-s + (−1.31 − 0.760i)7-s + (−0.0747 + 0.997i)9-s + (0.294 + 0.955i)12-s + (−0.562 − 1.43i)13-s + (0.623 + 0.781i)16-s + (1.60 − 0.120i)19-s + (−0.338 − 1.48i)21-s + (−0.955 − 0.294i)25-s + (−0.781 + 0.623i)27-s + (−0.856 − 1.25i)28-s + (−1.71 + 0.529i)31-s + (−0.5 + 0.866i)36-s + (1.34 − 0.777i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.790 - 0.613i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.790 - 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29427 + 0.443280i\)
\(L(\frac12)\) \(\approx\) \(1.29427 + 0.443280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.17 - 1.26i)T \)
43 \( 1 + (4.17 - 5.05i)T \)
good2 \( 1 + (-1.80 - 0.867i)T^{2} \)
5 \( 1 + (4.77 + 1.47i)T^{2} \)
7 \( 1 + (3.48 + 2.01i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (2.02 + 5.17i)T + (-9.52 + 8.84i)T^{2} \)
17 \( 1 + (-16.2 + 5.01i)T^{2} \)
19 \( 1 + (-7.00 + 0.525i)T + (18.7 - 2.83i)T^{2} \)
23 \( 1 + (-8.40 - 21.4i)T^{2} \)
29 \( 1 + (2.16 + 28.9i)T^{2} \)
31 \( 1 + (9.55 - 2.94i)T + (25.6 - 17.4i)T^{2} \)
37 \( 1 + (-8.18 + 4.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (36.9 + 17.7i)T^{2} \)
47 \( 1 + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (38.8 + 36.0i)T^{2} \)
59 \( 1 + (13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.0494 + 0.160i)T + (-50.4 - 34.3i)T^{2} \)
67 \( 1 + (-0.677 - 9.04i)T + (-66.2 + 9.98i)T^{2} \)
71 \( 1 + (25.9 - 66.0i)T^{2} \)
73 \( 1 + (-9.58 + 3.76i)T + (53.5 - 49.6i)T^{2} \)
79 \( 1 + (8.01 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.20 + 82.7i)T^{2} \)
89 \( 1 + (6.65 - 88.7i)T^{2} \)
97 \( 1 + (-1.89 + 0.913i)T + (60.4 - 75.8i)T^{2} \)
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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

−13.34102562385011410026258719501, −12.60303015982789069277482634336, −11.20025209532128017202194528678, −10.19023287540792886423902801212, −9.524954382650215631223514682236, −7.910165438178694712748635895769, −7.18918001860858559868893461940, −5.61907906046991385951971237796, −3.69270885002522305701048400710, −2.86538895493128751099249706914, 2.05637415020490042702307803583, 3.32362432360819950994314580463, 5.77074752167220132724812025851, 6.73058434157922760624450403512, 7.57768008164834770423016148427, 9.289667743761446137277717586496, 9.708975139720155349209101870928, 11.55568470311011529726186419725, 12.11710145289103643543711561160, 13.21913890958599569071865010945

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