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Properties

Label 2-471-157.144-c1-0-3
Degree $2$
Conductor $471$
Sign $-0.999 - 0.0266i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21·2-s + (−0.5 + 0.866i)3-s − 0.517·4-s + (−1.98 + 3.44i)5-s + (0.608 − 1.05i)6-s + 2.72·7-s + 3.06·8-s + (−0.499 − 0.866i)9-s + (2.42 − 4.19i)10-s + (−1.10 + 1.91i)11-s + (0.258 − 0.448i)12-s + (3.38 + 5.86i)13-s − 3.32·14-s + (−1.98 − 3.44i)15-s − 2.69·16-s + (−0.929 + 1.61i)17-s + ⋯
L(s)  = 1  − 0.860·2-s + (−0.288 + 0.499i)3-s − 0.258·4-s + (−0.889 + 1.54i)5-s + (0.248 − 0.430i)6-s + 1.03·7-s + 1.08·8-s + (−0.166 − 0.288i)9-s + (0.765 − 1.32i)10-s + (−0.332 + 0.576i)11-s + (0.0747 − 0.129i)12-s + (0.939 + 1.62i)13-s − 0.887·14-s + (−0.513 − 0.889i)15-s − 0.673·16-s + (−0.225 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0266i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0266i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.999 - 0.0266i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ -0.999 - 0.0266i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00639712 + 0.480175i\)
\(L(\frac12)\) \(\approx\) \(0.00639712 + 0.480175i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
157 \( 1 + (9.18 + 8.52i)T \)
good2 \( 1 + 1.21T + 2T^{2} \)
5 \( 1 + (1.98 - 3.44i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.72T + 7T^{2} \)
11 \( 1 + (1.10 - 1.91i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.38 - 5.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.929 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0693 + 0.120i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.480T + 23T^{2} \)
29 \( 1 + 9.21T + 29T^{2} \)
31 \( 1 + (5.29 + 9.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.12 - 3.68i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + (1.05 + 1.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.333 + 0.577i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.15 - 7.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4.30T + 59T^{2} \)
61 \( 1 + (3.04 - 5.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 1.96T + 67T^{2} \)
71 \( 1 + (-3.09 - 5.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.613 + 1.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.72T + 79T^{2} \)
83 \( 1 + (4.51 + 7.82i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.69 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.92 + 6.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.11088718326515424569265771189, −10.78626357317314553777745979339, −9.680925656089956379139277868611, −8.860539148641760697757779527715, −7.75206534307417025127507873474, −7.27960449949516925023380481781, −6.02214049989237324841704211905, −4.40029736138954528926457989767, −3.91147341560603115706012027073, −1.99361617848494566599805849889, 0.45853859470766273731599865823, 1.41884463721347897257969615277, 3.79192455829412120639773745010, 5.00661837162736384676740959389, 5.55002593847313532553788802713, 7.55919291273929544066332728118, 7.992674974879442735917875517006, 8.587839859777008441937201562632, 9.329111764473960960760113280750, 10.92780094321281885289074285419

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