Properties

Label 2-429-33.32-c1-0-16
Degree $2$
Conductor $429$
Sign $0.806 + 0.590i$
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  − 0.791·2-s + (−0.917 − 1.46i)3-s − 1.37·4-s + 1.17i·5-s + (0.726 + 1.16i)6-s − 0.422i·7-s + 2.67·8-s + (−1.31 + 2.69i)9-s − 0.931i·10-s + (1.23 + 3.07i)11-s + (1.25 + 2.01i)12-s i·13-s + 0.334i·14-s + (1.72 − 1.07i)15-s + 0.632·16-s + 0.541·17-s + ⋯
L(s)  = 1  − 0.559·2-s + (−0.529 − 0.848i)3-s − 0.686·4-s + 0.526i·5-s + (0.296 + 0.474i)6-s − 0.159i·7-s + 0.944·8-s + (−0.439 + 0.898i)9-s − 0.294i·10-s + (0.371 + 0.928i)11-s + (0.363 + 0.582i)12-s − 0.277i·13-s + 0.0893i·14-s + (0.446 − 0.278i)15-s + 0.158·16-s + 0.131·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $0.806 + 0.590i$
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: \((2,\ 429,\ (\ :1/2),\ 0.806 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681294 - 0.222724i\)
\(L(\frac12)\) \(\approx\) \(0.681294 - 0.222724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.917 + 1.46i)T \)
11 \( 1 + (-1.23 - 3.07i)T \)
13 \( 1 + iT \)
good2 \( 1 + 0.791T + 2T^{2} \)
5 \( 1 - 1.17iT - 5T^{2} \)
7 \( 1 + 0.422iT - 7T^{2} \)
17 \( 1 - 0.541T + 17T^{2} \)
19 \( 1 + 6.79iT - 19T^{2} \)
23 \( 1 + 1.90iT - 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 1.41T + 31T^{2} \)
37 \( 1 + 6.19T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 9.30iT - 43T^{2} \)
47 \( 1 - 1.34iT - 47T^{2} \)
53 \( 1 + 4.75iT - 53T^{2} \)
59 \( 1 - 7.23iT - 59T^{2} \)
61 \( 1 - 12.4iT - 61T^{2} \)
67 \( 1 - 9.42T + 67T^{2} \)
71 \( 1 + 6.54iT - 71T^{2} \)
73 \( 1 - 4.36iT - 73T^{2} \)
79 \( 1 - 4.73iT - 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 12.0iT - 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−10.82860861830764791388312149508, −10.34937115527375102272349730924, −9.174880314138733627782797335829, −8.339451164526446294657671540569, −7.22093272339407309159815213868, −6.77684673613818621700604297799, −5.31903978141395995635514095205, −4.37008701480639886053784034394, −2.54046432842201552618891076744, −0.868633113511815691854679942145, 1.00171938557520051869003960367, 3.48358209011616882870216856842, 4.47548900129254172116961557470, 5.39937107274513140475051407660, 6.36573148291166670596513888172, 7.989275650618806220153978475637, 8.736681463234413746200446776805, 9.413282481582685505836540863629, 10.24218969362889915608222036969, 11.00471233936843590888510053518

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