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Properties

Label 2-471-471.425-c0-0-0
Degree $2$
Conductor $471$
Sign $0.968 - 0.250i$
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  + (−0.970 − 0.239i)3-s + (0.970 − 0.239i)4-s + (−0.447 + 1.81i)7-s + (0.885 + 0.464i)9-s − 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (−0.688 − 1.81i)19-s + (0.869 − 1.65i)21-s + (−0.120 + 0.992i)25-s + (−0.748 − 0.663i)27-s + 1.87i·28-s + (−0.136 + 0.198i)31-s + (0.970 + 0.239i)36-s + (−0.530 − 0.470i)37-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)3-s + (0.970 − 0.239i)4-s + (−0.447 + 1.81i)7-s + (0.885 + 0.464i)9-s − 12-s + 1.13·13-s + (0.885 − 0.464i)16-s + (−0.688 − 1.81i)19-s + (0.869 − 1.65i)21-s + (−0.120 + 0.992i)25-s + (−0.748 − 0.663i)27-s + 1.87i·28-s + (−0.136 + 0.198i)31-s + (0.970 + 0.239i)36-s + (−0.530 − 0.470i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.968 - 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7971778630\)
\(L(\frac12)\) \(\approx\) \(0.7971778630\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.970 + 0.239i)T \)
157 \( 1 + (0.970 + 0.239i)T \)
good2 \( 1 + (-0.970 + 0.239i)T^{2} \)
5 \( 1 + (0.120 - 0.992i)T^{2} \)
7 \( 1 + (0.447 - 1.81i)T + (-0.885 - 0.464i)T^{2} \)
11 \( 1 + (0.970 + 0.239i)T^{2} \)
13 \( 1 - 1.13T + T^{2} \)
17 \( 1 + (-0.120 + 0.992i)T^{2} \)
19 \( 1 + (0.688 + 1.81i)T + (-0.748 + 0.663i)T^{2} \)
23 \( 1 + (0.568 - 0.822i)T^{2} \)
29 \( 1 + (0.120 + 0.992i)T^{2} \)
31 \( 1 + (0.136 - 0.198i)T + (-0.354 - 0.935i)T^{2} \)
37 \( 1 + (0.530 + 0.470i)T + (0.120 + 0.992i)T^{2} \)
41 \( 1 + (0.568 + 0.822i)T^{2} \)
43 \( 1 + (-0.222 - 0.902i)T + (-0.885 + 0.464i)T^{2} \)
47 \( 1 + (-0.120 + 0.992i)T^{2} \)
53 \( 1 + (-0.748 + 0.663i)T^{2} \)
59 \( 1 + (-0.748 - 0.663i)T^{2} \)
61 \( 1 + (1.85 - 0.704i)T + (0.748 - 0.663i)T^{2} \)
67 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
71 \( 1 + (0.748 - 0.663i)T^{2} \)
73 \( 1 + (-0.317 + 1.28i)T + (-0.885 - 0.464i)T^{2} \)
79 \( 1 + (0.475 + 0.0576i)T + (0.970 + 0.239i)T^{2} \)
83 \( 1 + (-0.354 - 0.935i)T^{2} \)
89 \( 1 + (0.354 + 0.935i)T^{2} \)
97 \( 1 + (0.879 + 0.992i)T + (-0.120 + 0.992i)T^{2} \)
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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

−11.21689545648417113895162328902, −10.81972989723655891182371667790, −9.491143014675283277086517302930, −8.687250317412093912587324221800, −7.34604523536642366063098379663, −6.34542951953910081896716755563, −5.92498231425277697886588318472, −4.94138799186438628148715463436, −3.03926560193354077143760916603, −1.83300945254946189663274826703, 1.40502878910738916859716162524, 3.58290765972168633758768008045, 4.19691073565468437213862560661, 5.89041943711187265480500841063, 6.51290110524052137179693837414, 7.31790784812170734288298966647, 8.255784268272680006240676217744, 9.957161880855415406552451571927, 10.50033531851318826995470632503, 10.97030275418508816456599625215

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