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Properties

Label 2-129-129.20-c1-0-6
Degree $2$
Conductor $129$
Sign $0.771 + 0.635i$
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  + (−1.61 + 0.632i)3-s + (0.445 − 1.94i)4-s + (3.67 − 2.12i)7-s + (2.19 − 2.04i)9-s + (0.516 + 3.42i)12-s + (0.108 + 0.0737i)13-s + (−3.60 − 1.73i)16-s + (−2.78 + 3.00i)19-s + (−4.58 + 5.75i)21-s + (4.94 + 0.745i)25-s + (−2.25 + 4.68i)27-s + (−2.50 − 8.11i)28-s + (−11.0 + 1.65i)31-s + (−3.00 − 5.19i)36-s + (10.2 + 5.94i)37-s + ⋯
L(s)  = 1  + (−0.930 + 0.365i)3-s + (0.222 − 0.974i)4-s + (1.39 − 0.802i)7-s + (0.733 − 0.680i)9-s + (0.149 + 0.988i)12-s + (0.0300 + 0.0204i)13-s + (−0.900 − 0.433i)16-s + (−0.638 + 0.688i)19-s + (−1.00 + 1.25i)21-s + (0.988 + 0.149i)25-s + (−0.433 + 0.900i)27-s + (−0.473 − 1.53i)28-s + (−1.97 + 0.298i)31-s + (−0.5 − 0.866i)36-s + (1.69 + 0.976i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.771 + 0.635i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.771 + 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.894978 - 0.321178i\)
\(L(\frac12)\) \(\approx\) \(0.894978 - 0.321178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.61 - 0.632i)T \)
43 \( 1 + (-1.56 - 6.36i)T \)
good2 \( 1 + (-0.445 + 1.94i)T^{2} \)
5 \( 1 + (-4.94 - 0.745i)T^{2} \)
7 \( 1 + (-3.67 + 2.12i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.108 - 0.0737i)T + (4.74 + 12.1i)T^{2} \)
17 \( 1 + (16.8 - 2.53i)T^{2} \)
19 \( 1 + (2.78 - 3.00i)T + (-1.41 - 18.9i)T^{2} \)
23 \( 1 + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (11.0 - 1.65i)T + (29.6 - 9.13i)T^{2} \)
37 \( 1 + (-10.2 - 5.94i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.12 - 39.9i)T^{2} \)
47 \( 1 + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-36.7 - 46.1i)T^{2} \)
61 \( 1 + (1.80 - 11.9i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (8.26 + 7.66i)T + (5.00 + 66.8i)T^{2} \)
71 \( 1 + (58.6 - 39.9i)T^{2} \)
73 \( 1 + (-8.55 + 12.5i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (1.95 + 3.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (60.8 - 56.4i)T^{2} \)
89 \( 1 + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-4.14 - 18.1i)T + (-87.3 + 42.0i)T^{2} \)
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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

−13.23314616624964374815766011319, −11.86072139330858423381094797468, −10.88410061628725627479084812932, −10.57585138062117053879877946289, −9.278976324435664357826480299444, −7.65923031023510203264535431752, −6.43019173241405053941142800051, −5.25093415294028437129137728562, −4.33394898539464610810077107119, −1.37785678583644518653878929374, 2.19150060445341984879710707734, 4.40523053677783783410743893667, 5.58390784141848461678264615250, 7.00786962861932774053064487197, 7.974438496546948555630717233380, 8.997426296479166909635413436241, 10.95647626181091171212166300503, 11.34142284361798597232095066866, 12.39612256657178747913310384390, 13.01371045725235070191618276499

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