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Properties

Label 2-471-471.422-c0-0-0
Degree $2$
Conductor $471$
Sign $0.815 + 0.579i$
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  + (0.120 − 0.992i)3-s + (0.120 + 0.992i)4-s + (0.136 − 1.12i)7-s + (−0.970 − 0.239i)9-s + 12-s + 1.77·13-s + (−0.970 + 0.239i)16-s + (0.136 − 0.198i)19-s + (−1.10 − 0.271i)21-s + (−0.748 + 0.663i)25-s + (−0.354 + 0.935i)27-s + 1.13·28-s + (−1.32 + 0.695i)31-s + (0.120 − 0.992i)36-s + (−0.402 + 1.06i)37-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)3-s + (0.120 + 0.992i)4-s + (0.136 − 1.12i)7-s + (−0.970 − 0.239i)9-s + 12-s + 1.77·13-s + (−0.970 + 0.239i)16-s + (0.136 − 0.198i)19-s + (−1.10 − 0.271i)21-s + (−0.748 + 0.663i)25-s + (−0.354 + 0.935i)27-s + 1.13·28-s + (−1.32 + 0.695i)31-s + (0.120 − 0.992i)36-s + (−0.402 + 1.06i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.815 + 0.579i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (422, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.815 + 0.579i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9293428860\)
\(L(\frac12)\) \(\approx\) \(0.9293428860\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.120 + 0.992i)T \)
157 \( 1 + (-0.120 + 0.992i)T \)
good2 \( 1 + (-0.120 - 0.992i)T^{2} \)
5 \( 1 + (0.748 - 0.663i)T^{2} \)
7 \( 1 + (-0.136 + 1.12i)T + (-0.970 - 0.239i)T^{2} \)
11 \( 1 + (-0.120 + 0.992i)T^{2} \)
13 \( 1 - 1.77T + T^{2} \)
17 \( 1 + (0.748 - 0.663i)T^{2} \)
19 \( 1 + (-0.136 + 0.198i)T + (-0.354 - 0.935i)T^{2} \)
23 \( 1 + (-0.885 + 0.464i)T^{2} \)
29 \( 1 + (0.748 + 0.663i)T^{2} \)
31 \( 1 + (1.32 - 0.695i)T + (0.568 - 0.822i)T^{2} \)
37 \( 1 + (0.402 - 1.06i)T + (-0.748 - 0.663i)T^{2} \)
41 \( 1 + (-0.885 - 0.464i)T^{2} \)
43 \( 1 + (0.234 + 1.92i)T + (-0.970 + 0.239i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
53 \( 1 + (0.354 + 0.935i)T^{2} \)
59 \( 1 + (0.354 - 0.935i)T^{2} \)
61 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
67 \( 1 + (1.10 - 1.59i)T + (-0.354 - 0.935i)T^{2} \)
71 \( 1 + (0.354 + 0.935i)T^{2} \)
73 \( 1 + (0.0854 - 0.704i)T + (-0.970 - 0.239i)T^{2} \)
79 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (-0.568 + 0.822i)T^{2} \)
89 \( 1 + (-0.568 + 0.822i)T^{2} \)
97 \( 1 + (-0.251 - 0.663i)T + (-0.748 + 0.663i)T^{2} \)
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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

−11.29799333846941830525534411633, −10.54246198739403000658102994901, −8.954718312589985244108078579309, −8.360639209874502965966658420564, −7.37950844616356743063600078108, −6.89866334752924158743231165730, −5.71802211844036783427500538757, −4.01463493856269096500958938127, −3.23632756741473239182474687081, −1.55525295829488536586614228034, 2.00586936534561740416622083590, 3.48479237991336208007319161412, 4.71581306650420855973416250450, 5.82030335728334149463744765626, 6.13377589864957755074135634666, 7.973350367153251071545846562407, 8.982740447478387502594955068153, 9.414085821174788244951808671618, 10.52259316976143295684183513149, 11.12072396593811234911116401950

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