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Properties

Label 2-471-471.353-c0-0-0
Degree $2$
Conductor $471$
Sign $-0.978 - 0.204i$
Analytic cond. $0.235059$
Root an. cond. $0.484829$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.748 + 0.663i)3-s + (−0.748 − 0.663i)4-s + (−1.32 + 1.17i)7-s + (0.120 − 0.992i)9-s + 12-s − 1.94·13-s + (0.120 + 0.992i)16-s + (−1.32 + 0.695i)19-s + (0.213 − 1.75i)21-s + (−0.354 − 0.935i)25-s + (0.568 + 0.822i)27-s + 1.77·28-s + (0.688 − 0.169i)31-s + (−0.748 + 0.663i)36-s + (1.00 + 1.45i)37-s + ⋯
L(s)  = 1  + (−0.748 + 0.663i)3-s + (−0.748 − 0.663i)4-s + (−1.32 + 1.17i)7-s + (0.120 − 0.992i)9-s + 12-s − 1.94·13-s + (0.120 + 0.992i)16-s + (−1.32 + 0.695i)19-s + (0.213 − 1.75i)21-s + (−0.354 − 0.935i)25-s + (0.568 + 0.822i)27-s + 1.77·28-s + (0.688 − 0.169i)31-s + (−0.748 + 0.663i)36-s + (1.00 + 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1443483201\)
\(L(\frac12)\) \(\approx\) \(0.1443483201\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.748 - 0.663i)T \)
157 \( 1 + (0.748 - 0.663i)T \)
good2 \( 1 + (0.748 + 0.663i)T^{2} \)
5 \( 1 + (0.354 + 0.935i)T^{2} \)
7 \( 1 + (1.32 - 1.17i)T + (0.120 - 0.992i)T^{2} \)
11 \( 1 + (0.748 - 0.663i)T^{2} \)
13 \( 1 + 1.94T + T^{2} \)
17 \( 1 + (0.354 + 0.935i)T^{2} \)
19 \( 1 + (1.32 - 0.695i)T + (0.568 - 0.822i)T^{2} \)
23 \( 1 + (0.970 - 0.239i)T^{2} \)
29 \( 1 + (0.354 - 0.935i)T^{2} \)
31 \( 1 + (-0.688 + 0.169i)T + (0.885 - 0.464i)T^{2} \)
37 \( 1 + (-1.00 - 1.45i)T + (-0.354 + 0.935i)T^{2} \)
41 \( 1 + (0.970 + 0.239i)T^{2} \)
43 \( 1 + (0.180 + 0.159i)T + (0.120 + 0.992i)T^{2} \)
47 \( 1 + (0.354 + 0.935i)T^{2} \)
53 \( 1 + (-0.568 + 0.822i)T^{2} \)
59 \( 1 + (-0.568 - 0.822i)T^{2} \)
61 \( 1 + (0.627 - 0.329i)T + (0.568 - 0.822i)T^{2} \)
67 \( 1 + (-0.213 + 0.112i)T + (0.568 - 0.822i)T^{2} \)
71 \( 1 + (-0.568 + 0.822i)T^{2} \)
73 \( 1 + (0.850 - 0.753i)T + (0.120 - 0.992i)T^{2} \)
79 \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.885 + 0.464i)T^{2} \)
89 \( 1 + (-0.885 + 0.464i)T^{2} \)
97 \( 1 + (-0.645 + 0.935i)T + (-0.354 - 0.935i)T^{2} \)
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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

−11.80460167959114297430754339919, −10.37317767945637275971459791573, −9.881110164025946561632524509041, −9.362163220729651077350562334625, −8.342742267482738328061851264704, −6.59424558309441388569292647014, −5.99799331952825278377149352958, −5.06715287433456419864780091566, −4.17213889687454280202093783097, −2.62019536954239019517230392602, 0.19234774823382123903625285949, 2.67848405417592635962469192314, 4.09863442877233049513321713615, 4.97112379825379800146307653546, 6.35846499499508337186939578607, 7.23535062598875956508436069946, 7.72207492390799603985275092720, 9.179226542354563856435117946731, 9.942106685778154534731173661016, 10.77622351297403549148503668272

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