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Properties

Label 2-429-11.3-c1-0-2
Degree $2$
Conductor $429$
Sign $0.337 - 0.941i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.57 + 1.14i)2-s + (0.309 + 0.951i)3-s + (0.559 − 1.72i)4-s + (−3.22 − 2.34i)5-s + (−1.57 − 1.14i)6-s + (0.498 − 1.53i)7-s + (−0.113 − 0.349i)8-s + (−0.809 + 0.587i)9-s + 7.77·10-s + (−1.04 + 3.14i)11-s + 1.81·12-s + (0.809 − 0.587i)13-s + (0.973 + 2.99i)14-s + (1.23 − 3.78i)15-s + (3.51 + 2.55i)16-s + (4.70 + 3.41i)17-s + ⋯
L(s)  = 1  + (−1.11 + 0.811i)2-s + (0.178 + 0.549i)3-s + (0.279 − 0.861i)4-s + (−1.44 − 1.04i)5-s + (−0.644 − 0.468i)6-s + (0.188 − 0.580i)7-s + (−0.0401 − 0.123i)8-s + (−0.269 + 0.195i)9-s + 2.45·10-s + (−0.315 + 0.949i)11-s + 0.522·12-s + (0.224 − 0.163i)13-s + (0.260 + 0.800i)14-s + (0.317 − 0.977i)15-s + (0.878 + 0.637i)16-s + (1.14 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.337 - 0.941i$
Analytic conductor: \(3.42558\)
Root analytic conductor: \(1.85083\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 429,\ (\ :1/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493647 + 0.347421i\)
\(L(\frac12)\) \(\approx\) \(0.493647 + 0.347421i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (1.04 - 3.14i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (1.57 - 1.14i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.22 + 2.34i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.498 + 1.53i)T + (-5.66 - 4.11i)T^{2} \)
17 \( 1 + (-4.70 - 3.41i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.39 - 4.28i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.15T + 23T^{2} \)
29 \( 1 + (-1.71 + 5.29i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.48 + 3.26i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.58 + 4.86i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.92 + 5.93i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.458T + 43T^{2} \)
47 \( 1 + (-4.07 - 12.5i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.78 + 1.29i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.24 + 3.82i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.64 + 4.82i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 7.44T + 67T^{2} \)
71 \( 1 + (-8.99 - 6.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.679 - 2.09i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.90 - 6.46i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-7.24 - 5.26i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 5.31T + 89T^{2} \)
97 \( 1 + (2.19 - 1.59i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.07163901563582605012509040744, −10.11199709942940227481096953247, −9.394899948144767199323351889067, −8.310550912932567085051339388097, −7.917284618425708836446872526116, −7.26223808588802248791721162721, −5.70142753203068094524632161298, −4.42623003011112926051031005819, −3.67650830604892803676910115361, −0.951326401869411189730323502319, 0.803542756010205649332726716692, 2.79327004694028246269953553683, 3.23726185310264917139511748023, 5.20627224306733419844622103745, 6.74576074825640721644219833322, 7.59707260412513301348032839373, 8.360450222393840320260190516443, 9.006833460502038217977438459037, 10.31090111005109880748148047957, 11.02021387988486358240461906078

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