↑

Properties

Label 2-129-129.128-c1-0-9
Degree $2$
Conductor $129$
Sign $0.948 + 0.316i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.27·2-s + (1.36 − 1.06i)3-s − 0.363·4-s + 0.284·5-s + (1.75 − 1.35i)6-s + 3.33i·7-s − 3.02·8-s + (0.752 − 2.90i)9-s + 0.363·10-s − 3.20i·11-s + (−0.497 + 0.385i)12-s − 2.50·13-s + 4.26i·14-s + (0.389 − 0.301i)15-s − 3.14·16-s + 1.65i·17-s + ⋯
L(s)  = 1  + 0.904·2-s + (0.790 − 0.612i)3-s − 0.181·4-s + 0.127·5-s + (0.715 − 0.553i)6-s + 1.25i·7-s − 1.06·8-s + (0.250 − 0.968i)9-s + 0.114·10-s − 0.966i·11-s + (−0.143 + 0.111i)12-s − 0.694·13-s + 1.13i·14-s + (0.100 − 0.0777i)15-s − 0.785·16-s + 0.401i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(129\)    =    \(3 \cdot 43\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(1.03007\)
Root analytic conductor: \(1.01492\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{129} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 129,\ (\ :1/2),\ 0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72395 - 0.280478i\)
\(L(\frac12)\) \(\approx\) \(1.72395 - 0.280478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 + 1.06i)T \)
43 \( 1 + (3.64 + 5.45i)T \)
good2 \( 1 - 1.27T + 2T^{2} \)
5 \( 1 - 0.284T + 5T^{2} \)
7 \( 1 - 3.33iT - 7T^{2} \)
11 \( 1 + 3.20iT - 11T^{2} \)
13 \( 1 + 2.50T + 13T^{2} \)
17 \( 1 - 1.65iT - 17T^{2} \)
19 \( 1 - 6.22iT - 19T^{2} \)
23 \( 1 + 1.05iT - 23T^{2} \)
29 \( 1 - 4.19T + 29T^{2} \)
31 \( 1 - 7.64T + 31T^{2} \)
37 \( 1 + 0.770iT - 37T^{2} \)
41 \( 1 + 4.36iT - 41T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 + 5.80iT - 59T^{2} \)
61 \( 1 - 8.78iT - 61T^{2} \)
67 \( 1 - 2.14T + 67T^{2} \)
71 \( 1 - 5.54T + 71T^{2} \)
73 \( 1 + 5.31iT - 73T^{2} \)
79 \( 1 - 6.77T + 79T^{2} \)
83 \( 1 - 9.01iT - 83T^{2} \)
89 \( 1 + 17.8T + 89T^{2} \)
97 \( 1 - 2.45T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.46801358769813159357185597875, −12.17458531964022595014598036684, −12.07961759570969091041088917974, −9.968125121661696998912079350169, −8.824546459019844255007139734543, −8.149983309336682831456876258528, −6.35273620166791776606645973639, −5.48516400037804152942941117750, −3.75639711475773821110208409319, −2.48612534692354645278597477086, 2.86167885188078477429493673098, 4.31764763788153682476247219711, 4.86030187911481469174773834800, 6.82066670207291034113153001930, 8.002472563609429703945100799871, 9.475504369637006476729312958136, 10.01277951682933933726093729197, 11.42756274659242347794572083500, 12.78553999832992765004007118274, 13.61460426224889882569791593196

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