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Properties

Label 2-471-471.275-c0-0-0
Degree $2$
Conductor $471$
Sign $0.286 - 0.957i$
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 + (0.616 + 0.695i)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 + (−0.922 − 0.112i)21-s + (0.354 + 0.935i)25-s + (0.568 + 0.822i)27-s + 0.929i·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 + (0.616 + 0.695i)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 + (−0.922 − 0.112i)21-s + (0.354 + 0.935i)25-s + (0.568 + 0.822i)27-s + 0.929i·28-s + (−0.688 + 0.169i)31-s + (0.748 − 0.663i)36-s + (−1.00 − 1.45i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.286 - 0.957i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.286 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8146849161\)
\(L(\frac12)\) \(\approx\) \(0.8146849161\)
\(L(1)\) 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 \( 1 + (-0.616 - 0.695i)T + (-0.120 + 0.992i)T^{2} \)
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 \( 1 + (-1.31 + 1.48i)T + (-0.120 - 0.992i)T^{2} \)
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 \( 1 + (0.869 + 1.65i)T + (-0.568 + 0.822i)T^{2} \)
67 \( 1 + (-0.213 + 0.112i)T + (0.568 - 0.822i)T^{2} \)
71 \( 1 + (-0.568 + 0.822i)T^{2} \)
73 \( 1 + (-1.09 - 1.23i)T + (-0.120 + 0.992i)T^{2} \)
79 \( 1 + (-1.24 + 0.470i)T + (0.748 - 0.663i)T^{2} \)
83 \( 1 + (0.885 - 0.464i)T^{2} \)
89 \( 1 + (-0.885 + 0.464i)T^{2} \)
97 \( 1 + (1.35 + 0.935i)T + (0.354 + 0.935i)T^{2} \)
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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

−11.38711467200043670488618827458, −10.79784072019701919764111699863, −9.636583576883877359981823763823, −8.911056738334412502112948809126, −7.51200323278854923835411227915, −6.99849665518915172745560578410, −5.53507961240475400537482706671, −4.96492276990317470871971238029, −3.53179732482937637906481017514, −2.27542178212090314672870136528, 1.30791064858942819155350538781, 2.60784259718480062366760655297, 4.68429993364365632806734208243, 5.39944785717508200749289715492, 6.49834332305021853867777401309, 7.39624003888706505225307440483, 7.81359801828983040906511653395, 9.629883063659697125038922719563, 10.34138088920264668400240578005, 11.07020948306827296043334423003

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