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Properties

Label 2-471-157.156-c3-0-52
Degree $2$
Conductor $471$
Sign $-0.928 - 0.371i$
Analytic cond. $27.7898$
Root an. cond. $5.27161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.06i·2-s − 3·3-s − 17.6·4-s + 10.9i·5-s + 15.1i·6-s − 24.6i·7-s + 48.9i·8-s + 9·9-s + 55.6·10-s + 13.6·11-s + 52.9·12-s + 52.7·13-s − 125.·14-s − 32.9i·15-s + 106.·16-s + 21.6·17-s + ⋯
L(s)  = 1  − 1.79i·2-s − 0.577·3-s − 2.20·4-s + 0.982i·5-s + 1.03i·6-s − 1.33i·7-s + 2.16i·8-s + 0.333·9-s + 1.75·10-s + 0.373·11-s + 1.27·12-s + 1.12·13-s − 2.38·14-s − 0.567i·15-s + 1.66·16-s + 0.309·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.928 - 0.371i$
Analytic conductor: \(27.7898\)
Root analytic conductor: \(5.27161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :3/2),\ -0.928 - 0.371i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.196562675\)
\(L(\frac12)\) \(\approx\) \(1.196562675\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
157 \( 1 + (1.82e3 + 731. i)T \)
good2 \( 1 + 5.06iT - 8T^{2} \)
5 \( 1 - 10.9iT - 125T^{2} \)
7 \( 1 + 24.6iT - 343T^{2} \)
11 \( 1 - 13.6T + 1.33e3T^{2} \)
13 \( 1 - 52.7T + 2.19e3T^{2} \)
17 \( 1 - 21.6T + 4.91e3T^{2} \)
19 \( 1 - 94.5T + 6.85e3T^{2} \)
23 \( 1 + 192. iT - 1.21e4T^{2} \)
29 \( 1 - 72.7iT - 2.43e4T^{2} \)
31 \( 1 + 212.T + 2.97e4T^{2} \)
37 \( 1 - 61.2T + 5.06e4T^{2} \)
41 \( 1 - 306. iT - 6.89e4T^{2} \)
43 \( 1 + 301. iT - 7.95e4T^{2} \)
47 \( 1 + 469.T + 1.03e5T^{2} \)
53 \( 1 + 141. iT - 1.48e5T^{2} \)
59 \( 1 + 122. iT - 2.05e5T^{2} \)
61 \( 1 + 323. iT - 2.26e5T^{2} \)
67 \( 1 - 52.0T + 3.00e5T^{2} \)
71 \( 1 + 43.9T + 3.57e5T^{2} \)
73 \( 1 - 76.7iT - 3.89e5T^{2} \)
79 \( 1 + 229. iT - 4.93e5T^{2} \)
83 \( 1 + 1.19e3iT - 5.71e5T^{2} \)
89 \( 1 - 252.T + 7.04e5T^{2} \)
97 \( 1 + 1.83e3iT - 9.12e5T^{2} \)
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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

−10.40898180590549660101945327804, −9.819025844616567733673184495250, −8.630145590067550781784298128739, −7.30930581240627965194005167168, −6.35333267172564809586350791601, −4.84534364050528160375093544411, −3.80557605075116676334300987046, −3.13538199105103340992108558002, −1.53005061303771630869216027604, −0.50640046427667216807140888319, 1.21859645112222913962652293267, 3.75114198961332777934745125722, 5.11031934742265569079570214734, 5.54240746675255107911967675108, 6.24319530948964306234386373738, 7.42638682515654567607388413678, 8.296801424296021863314529194631, 9.129713274721625249718935902407, 9.538656466250450300752860258078, 11.30710183859737031291825989134

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