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Properties

Label 2-471-157.156-c3-0-6
Degree $2$
Conductor $471$
Sign $0.179 + 0.983i$
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  + 3.78i·2-s − 3·3-s − 6.34·4-s + 9.26i·5-s − 11.3i·6-s + 4.30i·7-s + 6.25i·8-s + 9·9-s − 35.0·10-s + 9.78·11-s + 19.0·12-s + 35.0·13-s − 16.3·14-s − 27.7i·15-s − 74.4·16-s − 102.·17-s + ⋯
L(s)  = 1  + 1.33i·2-s − 0.577·3-s − 0.793·4-s + 0.828i·5-s − 0.773i·6-s + 0.232i·7-s + 0.276i·8-s + 0.333·9-s − 1.10·10-s + 0.268·11-s + 0.458·12-s + 0.748·13-s − 0.311·14-s − 0.478i·15-s − 1.16·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.179 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.179 + 0.983i$
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: \((2,\ 471,\ (\ :3/2),\ 0.179 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6192356008\)
\(L(\frac12)\) \(\approx\) \(0.6192356008\)
\(L(\frac{5}{2})\) 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 \( 1 + (-353. - 1.93e3i)T \)
good2 \( 1 - 3.78iT - 8T^{2} \)
5 \( 1 - 9.26iT - 125T^{2} \)
7 \( 1 - 4.30iT - 343T^{2} \)
11 \( 1 - 9.78T + 1.33e3T^{2} \)
13 \( 1 - 35.0T + 2.19e3T^{2} \)
17 \( 1 + 102.T + 4.91e3T^{2} \)
19 \( 1 + 92.8T + 6.85e3T^{2} \)
23 \( 1 - 177. iT - 1.21e4T^{2} \)
29 \( 1 - 88.6iT - 2.43e4T^{2} \)
31 \( 1 + 25.9T + 2.97e4T^{2} \)
37 \( 1 + 315.T + 5.06e4T^{2} \)
41 \( 1 + 81.5iT - 6.89e4T^{2} \)
43 \( 1 + 301. iT - 7.95e4T^{2} \)
47 \( 1 - 342.T + 1.03e5T^{2} \)
53 \( 1 + 38.5iT - 1.48e5T^{2} \)
59 \( 1 + 590. iT - 2.05e5T^{2} \)
61 \( 1 + 739. iT - 2.26e5T^{2} \)
67 \( 1 + 490.T + 3.00e5T^{2} \)
71 \( 1 + 234.T + 3.57e5T^{2} \)
73 \( 1 + 39.4iT - 3.89e5T^{2} \)
79 \( 1 + 21.1iT - 4.93e5T^{2} \)
83 \( 1 - 775. iT - 5.71e5T^{2} \)
89 \( 1 - 198.T + 7.04e5T^{2} \)
97 \( 1 + 482. iT - 9.12e5T^{2} \)
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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

−11.04929223909426860156709284690, −10.68787659677969508183724315049, −9.166894105809815771730968604717, −8.497104905848598280139849674702, −7.24683526315391926974692448999, −6.71107402382879028123383151172, −5.98271132437516663481301524494, −5.03576502761496447262086537206, −3.75720336306873806280547371847, −2.05590755997765355076796102324, 0.21684915279340132626781927632, 1.29865722575105584672561534331, 2.50129868600939616936927211503, 4.12460640555810946554762467809, 4.55776823121981190272454534853, 6.10915041701210200801797926179, 6.95136333612722221927884813820, 8.636942672993910882123763684994, 9.025996893593027593918665347937, 10.43934372692123734571072128913

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