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Properties

Label 2-122-61.20-c1-0-6
Degree $2$
Conductor $122$
Sign $-0.999 + 0.00141i$
Analytic cond. $0.974174$
Root an. cond. $0.987002$
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  + (−0.309 − 0.951i)2-s + (−0.811 − 2.49i)3-s + (−0.809 + 0.587i)4-s + (−3.26 + 2.37i)5-s + (−2.12 + 1.54i)6-s + (1.31 − 4.03i)7-s + (0.809 + 0.587i)8-s + (−3.15 + 2.29i)9-s + (3.26 + 2.37i)10-s − 1.83·11-s + (2.12 + 1.54i)12-s − 1.17·13-s − 4.24·14-s + (8.58 + 6.23i)15-s + (0.309 − 0.951i)16-s + (3.80 − 2.76i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (−0.468 − 1.44i)3-s + (−0.404 + 0.293i)4-s + (−1.46 + 1.06i)5-s + (−0.867 + 0.630i)6-s + (0.495 − 1.52i)7-s + (0.286 + 0.207i)8-s + (−1.05 + 0.764i)9-s + (1.03 + 0.750i)10-s − 0.552·11-s + (0.613 + 0.445i)12-s − 0.326·13-s − 1.13·14-s + (2.21 + 1.60i)15-s + (0.0772 − 0.237i)16-s + (0.921 − 0.669i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00141i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(122\)    =    \(2 \cdot 61\)
Sign: $-0.999 + 0.00141i$
Analytic conductor: \(0.974174\)
Root analytic conductor: \(0.987002\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{122} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 122,\ (\ :1/2),\ -0.999 + 0.00141i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000368746 - 0.520319i\)
\(L(\frac12)\) \(\approx\) \(0.000368746 - 0.520319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (-4.26 - 6.54i)T \)
good3 \( 1 + (0.811 + 2.49i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (3.26 - 2.37i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.31 + 4.03i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + 1.83T + 11T^{2} \)
13 \( 1 + 1.17T + 13T^{2} \)
17 \( 1 + (-3.80 + 2.76i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.82 + 5.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.944 + 0.686i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 - 5.84T + 29T^{2} \)
31 \( 1 + (0.794 - 2.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.227 - 0.701i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.634 - 1.95i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-6.70 - 4.87i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + 2.80T + 47T^{2} \)
53 \( 1 + (-0.358 - 0.260i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.301 + 0.928i)T + (-47.7 + 34.6i)T^{2} \)
67 \( 1 + (-2.21 + 1.60i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.12 + 1.54i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.08 - 1.51i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-7.31 - 5.31i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.50 + 13.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (4.95 + 15.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.08 + 9.50i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.72410878081755018714924144599, −11.73412018919792465747694463212, −11.12218648038876770712337961510, −10.32960160511482844051983902484, −8.120000971782455339594997968329, −7.41152653611665516268678014454, −6.86684211784505944471333473602, −4.53395991926302055122041437935, −2.94455472574400333165867179439, −0.65550038516337245844380320587, 3.88202939860318515449185264566, 4.99686508869047436534433623592, 5.65957502052016711615696735460, 7.954362306884394054431999462632, 8.535004841166427495868001787437, 9.575300689428459924919698774752, 10.79778251416735993911171581676, 11.97671433215475479198158942802, 12.50723312980317562850405078859, 14.64556207912064042930396300250

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