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Properties

Label 2-129-129.11-c0-0-0
Degree $2$
Conductor $129$
Sign $0.988 + 0.150i$
Analytic cond. $0.0643793$
Root an. cond. $0.253730$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.623 − 0.781i)3-s + (−0.222 + 0.974i)4-s − 1.80·7-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)12-s + (0.400 − 0.193i)13-s + (−0.900 − 0.433i)16-s + (−0.277 + 1.21i)19-s + (−1.12 + 1.40i)21-s + (0.623 − 0.781i)25-s + (−0.900 − 0.433i)27-s + (0.400 − 1.75i)28-s + (0.777 + 0.974i)31-s + 36-s − 0.445·37-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)3-s + (−0.222 + 0.974i)4-s − 1.80·7-s + (−0.222 − 0.974i)9-s + (0.623 + 0.781i)12-s + (0.400 − 0.193i)13-s + (−0.900 − 0.433i)16-s + (−0.277 + 1.21i)19-s + (−1.12 + 1.40i)21-s + (0.623 − 0.781i)25-s + (−0.900 − 0.433i)27-s + (0.400 − 1.75i)28-s + (0.777 + 0.974i)31-s + 36-s − 0.445·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(0.0643793\)
Root analytic conductor: \(0.253730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :0),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6200503732\)
\(L(\frac12)\) \(\approx\) \(0.6200503732\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (0.222 - 0.974i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T^{2} \)
7 \( 1 + 1.80T + T^{2} \)
11 \( 1 + (0.900 - 0.433i)T^{2} \)
13 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.623 - 0.781i)T^{2} \)
19 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
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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

−13.36500631173665182626393085077, −12.57488235395403764659415280905, −12.12023276593579841919249826409, −10.26264333481996414389821421833, −9.115486154676504315771557643676, −8.277214629916687585408263023458, −7.07768067308750310483855913972, −6.20105180911059626308592776543, −3.78897603568362382235659278873, −2.86012454576471401070129813826, 2.88325771425224940394628159345, 4.33280687056394933560473687743, 5.78240941938146748824322621911, 6.91966399279496071088329594102, 8.810426837993992849677089173916, 9.486667543166512574391777251332, 10.23070536882503965117717150118, 11.23085784124744062990294406030, 13.02902074927801376082986676113, 13.55402544841751887749647923387

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