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Properties

Label 2-129-129.41-c0-0-0
Degree $2$
Conductor $129$
Sign $0.785 + 0.618i$
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  + (−0.900 − 0.433i)3-s + (0.623 − 0.781i)4-s − 0.445·7-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)12-s + (−0.277 + 1.21i)13-s + (−0.222 − 0.974i)16-s + (−1.12 + 1.40i)19-s + (0.400 + 0.193i)21-s + (−0.900 − 0.433i)25-s + (−0.222 − 0.974i)27-s + (−0.277 + 0.347i)28-s + (1.62 − 0.781i)31-s + 36-s + 1.24·37-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (0.623 − 0.781i)4-s − 0.445·7-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)12-s + (−0.277 + 1.21i)13-s + (−0.222 − 0.974i)16-s + (−1.12 + 1.40i)19-s + (0.400 + 0.193i)21-s + (−0.900 − 0.433i)25-s + (−0.222 − 0.974i)27-s + (−0.277 + 0.347i)28-s + (1.62 − 0.781i)31-s + 36-s + 1.24·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.785 + 0.618i$
Analytic conductor: \(0.0643793\)
Root analytic conductor: \(0.253730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :0),\ 0.785 + 0.618i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5257211961\)
\(L(\frac12)\) \(\approx\) \(0.5257211961\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
good2 \( 1 + (-0.623 + 0.781i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 + 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 + 0.445T + T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.623 - 0.781i)T^{2} \)
97 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
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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

−13.41807199030638410866842268875, −12.16964255257150350025442914663, −11.55308952868367101444810472382, −10.43414145918561282295680127022, −9.673103059722523245203088815217, −7.86604371135424953251934593978, −6.52759954990724590546404907857, −6.05275396843483118535108134506, −4.49690869591238448603946969061, −1.97541040425816302260244792610, 3.00762260050070000034605042139, 4.54089190991418433684781540652, 6.05708096117740400824353938781, 7.00950574033900644621929818794, 8.306216461760531779481888651709, 9.752134875357311223684814448228, 10.77170372107125401077770228487, 11.59809604384221456103979014060, 12.59276808374635779367479676678, 13.25553767430018582032677475130

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