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Properties

Label 2-429-33.32-c1-0-41
Degree $2$
Conductor $429$
Sign $-0.116 + 0.993i$
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  + 0.426·2-s + (1.61 − 0.620i)3-s − 1.81·4-s − 1.27i·5-s + (0.689 − 0.264i)6-s − 2.98i·7-s − 1.62·8-s + (2.22 − 2.00i)9-s − 0.545i·10-s + (−2.93 − 1.54i)11-s + (−2.94 + 1.12i)12-s + i·13-s − 1.27i·14-s + (−0.794 − 2.06i)15-s + 2.94·16-s − 3.38·17-s + ⋯
L(s)  = 1  + 0.301·2-s + (0.933 − 0.358i)3-s − 0.909·4-s − 0.572i·5-s + (0.281 − 0.108i)6-s − 1.12i·7-s − 0.575·8-s + (0.743 − 0.669i)9-s − 0.172i·10-s + (−0.885 − 0.465i)11-s + (−0.848 + 0.325i)12-s + 0.277i·13-s − 0.339i·14-s + (−0.205 − 0.534i)15-s + 0.735·16-s − 0.820·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.116 + 0.993i$
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: \((2,\ 429,\ (\ :1/2),\ -0.116 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04451 - 1.17464i\)
\(L(\frac12)\) \(\approx\) \(1.04451 - 1.17464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 + 0.620i)T \)
11 \( 1 + (2.93 + 1.54i)T \)
13 \( 1 - iT \)
good2 \( 1 - 0.426T + 2T^{2} \)
5 \( 1 + 1.27iT - 5T^{2} \)
7 \( 1 + 2.98iT - 7T^{2} \)
17 \( 1 + 3.38T + 17T^{2} \)
19 \( 1 - 3.53iT - 19T^{2} \)
23 \( 1 + 6.34iT - 23T^{2} \)
29 \( 1 - 8.58T + 29T^{2} \)
31 \( 1 - 0.512T + 31T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 - 0.348T + 41T^{2} \)
43 \( 1 + 8.37iT - 43T^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 - 1.62iT - 53T^{2} \)
59 \( 1 - 5.84iT - 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 - 5.63T + 67T^{2} \)
71 \( 1 + 1.70iT - 71T^{2} \)
73 \( 1 + 3.88iT - 73T^{2} \)
79 \( 1 + 6.66iT - 79T^{2} \)
83 \( 1 + 8.81T + 83T^{2} \)
89 \( 1 - 4.99iT - 89T^{2} \)
97 \( 1 + 9.51T + 97T^{2} \)
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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

−10.66080334065728266105729379043, −9.988247519096856140830615974315, −8.802098041014078112993637566880, −8.406676763087339531210974476277, −7.39765010438978537192642625894, −6.22532240218961245777437043501, −4.71745075626115189971662528739, −4.10807540466946681585688799172, −2.82612262222100051993733174180, −0.874735786634518065436994072052, 2.43296462791866144942000926261, 3.26067583255430725748611066414, 4.63848370296217509539148078878, 5.34936318521490255650058207970, 6.78628023534135894036433064313, 8.035905320049668014102654533047, 8.710223862276471781323354606561, 9.525944903911990794342906874489, 10.25484526410487566649262743018, 11.37767869080866435734945595403

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