↑

Properties

Label 2-429-33.32-c1-0-29
Degree $2$
Conductor $429$
Sign $0.785 - 0.619i$
Analytic cond. $3.42558$
Root an. cond. $1.85083$
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  + 1.88·2-s + (1.68 + 0.400i)3-s + 1.55·4-s + 4.10i·5-s + (3.17 + 0.754i)6-s − 3.01i·7-s − 0.847·8-s + (2.67 + 1.34i)9-s + 7.73i·10-s + (−2.60 − 2.05i)11-s + (2.61 + 0.620i)12-s − i·13-s − 5.68i·14-s + (−1.64 + 6.91i)15-s − 4.69·16-s + 6.56·17-s + ⋯
L(s)  = 1  + 1.33·2-s + (0.972 + 0.231i)3-s + 0.775·4-s + 1.83i·5-s + (1.29 + 0.308i)6-s − 1.14i·7-s − 0.299·8-s + (0.893 + 0.449i)9-s + 2.44i·10-s + (−0.783 − 0.620i)11-s + (0.754 + 0.179i)12-s − 0.277i·13-s − 1.52i·14-s + (−0.424 + 1.78i)15-s − 1.17·16-s + 1.59·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.785 - 0.619i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.785 - 0.619i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.15982 + 1.09597i\)
\(L(\frac12)\) \(\approx\) \(3.15982 + 1.09597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.400i)T \)
11 \( 1 + (2.60 + 2.05i)T \)
13 \( 1 + iT \)
good2 \( 1 - 1.88T + 2T^{2} \)
5 \( 1 - 4.10iT - 5T^{2} \)
7 \( 1 + 3.01iT - 7T^{2} \)
17 \( 1 - 6.56T + 17T^{2} \)
19 \( 1 - 1.38iT - 19T^{2} \)
23 \( 1 + 5.41iT - 23T^{2} \)
29 \( 1 + 6.07T + 29T^{2} \)
31 \( 1 - 6.13T + 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
41 \( 1 + 0.966T + 41T^{2} \)
43 \( 1 + 3.07iT - 43T^{2} \)
47 \( 1 - 2.62iT - 47T^{2} \)
53 \( 1 + 7.86iT - 53T^{2} \)
59 \( 1 - 3.17iT - 59T^{2} \)
61 \( 1 + 4.19iT - 61T^{2} \)
67 \( 1 + 0.278T + 67T^{2} \)
71 \( 1 - 1.72iT - 71T^{2} \)
73 \( 1 - 3.90iT - 73T^{2} \)
79 \( 1 - 6.67iT - 79T^{2} \)
83 \( 1 + 6.62T + 83T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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

−11.14923220092608954726384968178, −10.42364701389615745217557147606, −9.880462909050011383851725877050, −8.194063231723422015928358632446, −7.39293590166592517938959045498, −6.56907174757500639610253224727, −5.40745723470577540517301210934, −3.96228675458308956283413497604, −3.35585017998576263112324695813, −2.59740226336250688397886886304, 1.76744640262602276875531178633, 3.08225276981662619840099224459, 4.29582032158703397632995045352, 5.19451110344335896953608387372, 5.78926571973056780447267760852, 7.47785562562795819993350031990, 8.393618400918442650102781627937, 9.173139191607256005454769423889, 9.787792914301171727481833807558, 11.86731102748426330998696853131

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