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Properties

Label 2-471-157.156-c3-0-33
Degree $2$
Conductor $471$
Sign $-0.218 - 0.975i$
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  + 0.275i·2-s − 3·3-s + 7.92·4-s + 19.4i·5-s − 0.826i·6-s + 15.2i·7-s + 4.38i·8-s + 9·9-s − 5.36·10-s + 43.0·11-s − 23.7·12-s + 61.4·13-s − 4.20·14-s − 58.3i·15-s + 62.1·16-s − 27.8·17-s + ⋯
L(s)  = 1  + 0.0974i·2-s − 0.577·3-s + 0.990·4-s + 1.73i·5-s − 0.0562i·6-s + 0.823i·7-s + 0.193i·8-s + 0.333·9-s − 0.169·10-s + 1.17·11-s − 0.571·12-s + 1.31·13-s − 0.0802·14-s − 1.00i·15-s + 0.971·16-s − 0.396·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.218 - 0.975i$
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: \((2,\ 471,\ (\ :3/2),\ -0.218 - 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.427372469\)
\(L(\frac12)\) \(\approx\) \(2.427372469\)
\(L(\frac{5}{2})\) 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 \( 1 + (429. + 1.91e3i)T \)
good2 \( 1 - 0.275iT - 8T^{2} \)
5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 - 15.2iT - 343T^{2} \)
11 \( 1 - 43.0T + 1.33e3T^{2} \)
13 \( 1 - 61.4T + 2.19e3T^{2} \)
17 \( 1 + 27.8T + 4.91e3T^{2} \)
19 \( 1 - 113.T + 6.85e3T^{2} \)
23 \( 1 - 42.1iT - 1.21e4T^{2} \)
29 \( 1 + 88.4iT - 2.43e4T^{2} \)
31 \( 1 + 16.5T + 2.97e4T^{2} \)
37 \( 1 - 146.T + 5.06e4T^{2} \)
41 \( 1 + 86.5iT - 6.89e4T^{2} \)
43 \( 1 + 67.8iT - 7.95e4T^{2} \)
47 \( 1 + 366.T + 1.03e5T^{2} \)
53 \( 1 - 192. iT - 1.48e5T^{2} \)
59 \( 1 - 297. iT - 2.05e5T^{2} \)
61 \( 1 + 761. iT - 2.26e5T^{2} \)
67 \( 1 + 343.T + 3.00e5T^{2} \)
71 \( 1 + 781.T + 3.57e5T^{2} \)
73 \( 1 + 274. iT - 3.89e5T^{2} \)
79 \( 1 - 331. iT - 4.93e5T^{2} \)
83 \( 1 - 145. iT - 5.71e5T^{2} \)
89 \( 1 + 208.T + 7.04e5T^{2} \)
97 \( 1 - 1.51e3iT - 9.12e5T^{2} \)
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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

−11.14206423251529486238740785499, −10.22733056309376159928666350326, −9.223369527271803158986257911871, −7.82947719721888131222374442554, −6.90908725348676875787092173039, −6.29303113063980216152245071688, −5.70232904512778614895042216395, −3.74330075484024654799258317190, −2.84151407746202205513143432262, −1.56571801896830193667599012642, 0.967974445544750535246792019189, 1.41841723798480420688702064467, 3.59979031051789932901206056810, 4.54184999860262936629387467489, 5.69183971867927931110231421789, 6.53275646449781334071895291771, 7.54637701210699477438719430596, 8.600265509371304829842455704300, 9.462708965585910660709023855322, 10.47769993285090100411826676771

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