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Properties

Label 2-471-157.156-c3-0-29
Degree $2$
Conductor $471$
Sign $-0.793 - 0.608i$
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  + 3.18i·2-s − 3·3-s − 2.11·4-s + 2.38i·5-s − 9.54i·6-s + 14.3i·7-s + 18.7i·8-s + 9·9-s − 7.58·10-s + 46.5·11-s + 6.34·12-s + 44.4·13-s − 45.5·14-s − 7.15i·15-s − 76.4·16-s + 77.4·17-s + ⋯
L(s)  = 1  + 1.12i·2-s − 0.577·3-s − 0.264·4-s + 0.213i·5-s − 0.649i·6-s + 0.773i·7-s + 0.827i·8-s + 0.333·9-s − 0.239·10-s + 1.27·11-s + 0.152·12-s + 0.948·13-s − 0.869·14-s − 0.123i·15-s − 1.19·16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.793 - 0.608i$
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: \((2,\ 471,\ (\ :3/2),\ -0.793 - 0.608i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.053066071\)
\(L(\frac12)\) \(\approx\) \(2.053066071\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.56e3 + 1.19e3i)T \)
good2 \( 1 - 3.18iT - 8T^{2} \)
5 \( 1 - 2.38iT - 125T^{2} \)
7 \( 1 - 14.3iT - 343T^{2} \)
11 \( 1 - 46.5T + 1.33e3T^{2} \)
13 \( 1 - 44.4T + 2.19e3T^{2} \)
17 \( 1 - 77.4T + 4.91e3T^{2} \)
19 \( 1 - 43.7T + 6.85e3T^{2} \)
23 \( 1 + 134. iT - 1.21e4T^{2} \)
29 \( 1 - 12.1iT - 2.43e4T^{2} \)
31 \( 1 - 174.T + 2.97e4T^{2} \)
37 \( 1 + 384.T + 5.06e4T^{2} \)
41 \( 1 - 334. iT - 6.89e4T^{2} \)
43 \( 1 - 395. iT - 7.95e4T^{2} \)
47 \( 1 + 189.T + 1.03e5T^{2} \)
53 \( 1 + 228. iT - 1.48e5T^{2} \)
59 \( 1 - 139. iT - 2.05e5T^{2} \)
61 \( 1 - 278. iT - 2.26e5T^{2} \)
67 \( 1 - 209.T + 3.00e5T^{2} \)
71 \( 1 + 322.T + 3.57e5T^{2} \)
73 \( 1 + 41.0iT - 3.89e5T^{2} \)
79 \( 1 - 250. iT - 4.93e5T^{2} \)
83 \( 1 + 929. iT - 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 646. iT - 9.12e5T^{2} \)
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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

−11.11419574221604414512261748693, −10.00071398701539770644806865394, −8.850136301108506902135643837638, −8.215339466987299896958840100959, −6.96486716574867076382188126948, −6.36214296366245735025811726657, −5.65081882345287598364056065252, −4.59869413919255402073046819406, −3.05921615841465252523382618212, −1.32298479243343353816160582839, 0.859875694005620224243911553844, 1.51163796398893244614460133550, 3.40336614149089576767164578992, 3.98692033388265921639703412412, 5.38298302081348513641959150364, 6.58749276444190787651466982217, 7.31601695394942026732749603339, 8.757385535027329029099863216192, 9.706860294131958957544643777519, 10.45307382010219767624925924581

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