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Properties

Label 2-471-157.156-c3-0-54
Degree $2$
Conductor $471$
Sign $0.539 + 0.842i$
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  − 0.500i·2-s − 3·3-s + 7.74·4-s + 10.0i·5-s + 1.50i·6-s − 27.3i·7-s − 7.88i·8-s + 9·9-s + 5.02·10-s + 72.6·11-s − 23.2·12-s − 24.6·13-s − 13.6·14-s − 30.1i·15-s + 58.0·16-s − 31.9·17-s + ⋯
L(s)  = 1  − 0.176i·2-s − 0.577·3-s + 0.968·4-s + 0.898i·5-s + 0.102i·6-s − 1.47i·7-s − 0.348i·8-s + 0.333·9-s + 0.159·10-s + 1.99·11-s − 0.559·12-s − 0.525·13-s − 0.260·14-s − 0.518i·15-s + 0.907·16-s − 0.455·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.539 + 0.842i$
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: \((2,\ 471,\ (\ :3/2),\ 0.539 + 0.842i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.108197011\)
\(L(\frac12)\) \(\approx\) \(2.108197011\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.06e3 - 1.65e3i)T \)
good2 \( 1 + 0.500iT - 8T^{2} \)
5 \( 1 - 10.0iT - 125T^{2} \)
7 \( 1 + 27.3iT - 343T^{2} \)
11 \( 1 - 72.6T + 1.33e3T^{2} \)
13 \( 1 + 24.6T + 2.19e3T^{2} \)
17 \( 1 + 31.9T + 4.91e3T^{2} \)
19 \( 1 + 120.T + 6.85e3T^{2} \)
23 \( 1 + 159. iT - 1.21e4T^{2} \)
29 \( 1 - 160. iT - 2.43e4T^{2} \)
31 \( 1 - 78.7T + 2.97e4T^{2} \)
37 \( 1 + 65.0T + 5.06e4T^{2} \)
41 \( 1 + 480. iT - 6.89e4T^{2} \)
43 \( 1 - 27.9iT - 7.95e4T^{2} \)
47 \( 1 - 478.T + 1.03e5T^{2} \)
53 \( 1 + 188. iT - 1.48e5T^{2} \)
59 \( 1 - 204. iT - 2.05e5T^{2} \)
61 \( 1 + 325. iT - 2.26e5T^{2} \)
67 \( 1 - 904.T + 3.00e5T^{2} \)
71 \( 1 - 546.T + 3.57e5T^{2} \)
73 \( 1 - 324. iT - 3.89e5T^{2} \)
79 \( 1 + 291. iT - 4.93e5T^{2} \)
83 \( 1 + 1.10e3iT - 5.71e5T^{2} \)
89 \( 1 + 776.T + 7.04e5T^{2} \)
97 \( 1 + 1.64e3iT - 9.12e5T^{2} \)
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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

−10.73594936545114485774352416374, −10.02890141782894664605145184458, −8.680135214789452730842747768631, −7.05467396233885833678414058553, −6.93300940774855144800989571206, −6.22558536406804832193118475593, −4.40433962246260374247552062095, −3.61976776402693343690886056264, −2.11624470686566654220492377863, −0.77051925348557428628818499976, 1.30755831475968210077204307805, 2.37223981343650547630144655924, 4.08276388666102779139398866014, 5.26494360440980740081622624760, 6.16772994194857799207103376692, 6.72539357810222278388349572917, 8.109328028574771520838667353875, 8.996274269703815487730260085035, 9.666289758769924106515580421793, 11.08064106477895332537251886431

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