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Properties

Label 2-471-157.156-c3-0-5
Degree $2$
Conductor $471$
Sign $0.982 - 0.184i$
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  + 5.39i·2-s − 3·3-s − 21.1·4-s + 16.5i·5-s − 16.1i·6-s + 25.8i·7-s − 70.7i·8-s + 9·9-s − 89.1·10-s + 42.1·11-s + 63.3·12-s − 86.3·13-s − 139.·14-s − 49.5i·15-s + 213.·16-s − 138.·17-s + ⋯
L(s)  = 1  + 1.90i·2-s − 0.577·3-s − 2.63·4-s + 1.47i·5-s − 1.10i·6-s + 1.39i·7-s − 3.12i·8-s + 0.333·9-s − 2.81·10-s + 1.15·11-s + 1.52·12-s − 1.84·13-s − 2.66·14-s − 0.852i·15-s + 3.32·16-s − 1.97·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.982 - 0.184i$
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: \((2,\ 471,\ (\ :3/2),\ 0.982 - 0.184i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4233259134\)
\(L(\frac12)\) \(\approx\) \(0.4233259134\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.93e3 + 363. i)T \)
good2 \( 1 - 5.39iT - 8T^{2} \)
5 \( 1 - 16.5iT - 125T^{2} \)
7 \( 1 - 25.8iT - 343T^{2} \)
11 \( 1 - 42.1T + 1.33e3T^{2} \)
13 \( 1 + 86.3T + 2.19e3T^{2} \)
17 \( 1 + 138.T + 4.91e3T^{2} \)
19 \( 1 - 58.3T + 6.85e3T^{2} \)
23 \( 1 + 80.8iT - 1.21e4T^{2} \)
29 \( 1 - 128. iT - 2.43e4T^{2} \)
31 \( 1 - 199.T + 2.97e4T^{2} \)
37 \( 1 + 193.T + 5.06e4T^{2} \)
41 \( 1 + 79.7iT - 6.89e4T^{2} \)
43 \( 1 - 380. iT - 7.95e4T^{2} \)
47 \( 1 - 219.T + 1.03e5T^{2} \)
53 \( 1 + 69.8iT - 1.48e5T^{2} \)
59 \( 1 - 755. iT - 2.05e5T^{2} \)
61 \( 1 - 529. iT - 2.26e5T^{2} \)
67 \( 1 - 769.T + 3.00e5T^{2} \)
71 \( 1 + 312.T + 3.57e5T^{2} \)
73 \( 1 + 357. iT - 3.89e5T^{2} \)
79 \( 1 + 395. iT - 4.93e5T^{2} \)
83 \( 1 + 390. iT - 5.71e5T^{2} \)
89 \( 1 - 866.T + 7.04e5T^{2} \)
97 \( 1 - 718. iT - 9.12e5T^{2} \)
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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

−11.67497456973216504162752798520, −10.36922317413143004493413038047, −9.380777958083219013486117926342, −8.761007657955755785426629117938, −7.44677320919867637646149517735, −6.72144136028969550773827809546, −6.34121834732997950318915934257, −5.25582712637930429409314192656, −4.35558179973390121830304543280, −2.63950090596566983884804285868, 0.19010107275823740251277545152, 0.939247237029545033343594542967, 2.05181187786858454270004096826, 3.87212619171328003018352165000, 4.52119276449626132551511087257, 5.11218965814320999341656338574, 6.96298432343156197420871227848, 8.278679775576959439944802418069, 9.394567152615737672877340709433, 9.697349183996974974776284188596

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