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Properties

Label 2-471-157.156-c3-0-74
Degree $2$
Conductor $471$
Sign $0.0735 - 0.997i$
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  − 2.63i·2-s − 3·3-s + 1.06·4-s − 20.9i·5-s + 7.89i·6-s + 20.4i·7-s − 23.8i·8-s + 9·9-s − 55.1·10-s − 26.5·11-s − 3.20·12-s − 27.3·13-s + 53.7·14-s + 62.8i·15-s − 54.3·16-s − 12.4·17-s + ⋯
L(s)  = 1  − 0.930i·2-s − 0.577·3-s + 0.133·4-s − 1.87i·5-s + 0.537i·6-s + 1.10i·7-s − 1.05i·8-s + 0.333·9-s − 1.74·10-s − 0.728·11-s − 0.0771·12-s − 0.583·13-s + 1.02·14-s + 1.08i·15-s − 0.848·16-s − 0.177·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0735 - 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0735 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.0735 - 0.997i$
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: \((2,\ 471,\ (\ :3/2),\ 0.0735 - 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3257608463\)
\(L(\frac12)\) \(\approx\) \(0.3257608463\)
\(L(\frac{5}{2})\) 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 \( 1 + (-144. + 1.96e3i)T \)
good2 \( 1 + 2.63iT - 8T^{2} \)
5 \( 1 + 20.9iT - 125T^{2} \)
7 \( 1 - 20.4iT - 343T^{2} \)
11 \( 1 + 26.5T + 1.33e3T^{2} \)
13 \( 1 + 27.3T + 2.19e3T^{2} \)
17 \( 1 + 12.4T + 4.91e3T^{2} \)
19 \( 1 - 25.3T + 6.85e3T^{2} \)
23 \( 1 - 63.4iT - 1.21e4T^{2} \)
29 \( 1 + 1.25iT - 2.43e4T^{2} \)
31 \( 1 + 142.T + 2.97e4T^{2} \)
37 \( 1 + 128.T + 5.06e4T^{2} \)
41 \( 1 + 218. iT - 6.89e4T^{2} \)
43 \( 1 + 117. iT - 7.95e4T^{2} \)
47 \( 1 + 48.1T + 1.03e5T^{2} \)
53 \( 1 - 505. iT - 1.48e5T^{2} \)
59 \( 1 - 105. iT - 2.05e5T^{2} \)
61 \( 1 - 7.96iT - 2.26e5T^{2} \)
67 \( 1 - 573.T + 3.00e5T^{2} \)
71 \( 1 - 236.T + 3.57e5T^{2} \)
73 \( 1 - 1.20e3iT - 3.89e5T^{2} \)
79 \( 1 - 871. iT - 4.93e5T^{2} \)
83 \( 1 - 218. iT - 5.71e5T^{2} \)
89 \( 1 + 822.T + 7.04e5T^{2} \)
97 \( 1 + 360. iT - 9.12e5T^{2} \)
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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

−9.891974627995772815354084835688, −9.261873163286915063630935623845, −8.373614034086029457455634157937, −7.20781664997468856798735488890, −5.67165062176614939777721930606, −5.21092772148990625793195613003, −4.03220725635830608695242447723, −2.43646182643876058033712485545, −1.38458096750135390096046359144, −0.10590551700025503919017934531, 2.24098610153085981608196158738, 3.42363179606031062728451886597, 4.89492331932795578991343796641, 6.04738680196564773235207990005, 6.82722525697562978472499074380, 7.30290254117542324210022159645, 8.005356005812429616902053154035, 9.804032634762542598661468951132, 10.62325316885611471128357587181, 10.98426199794966219708119665522

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