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Properties

Label 2-471-471.221-c0-0-0
Degree $2$
Conductor $471$
Sign $0.522 + 0.852i$
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  + (0.568 − 0.822i)3-s + (−0.568 − 0.822i)4-s + (1.63 + 1.12i)7-s + (−0.354 − 0.935i)9-s − 12-s − 1.49·13-s + (−0.354 + 0.935i)16-s + (−0.136 − 1.12i)19-s + (1.85 − 0.704i)21-s + (−0.885 + 0.464i)25-s + (−0.970 − 0.239i)27-s − 1.98i·28-s + (1.32 + 1.17i)31-s + (−0.568 + 0.822i)36-s + (0.234 + 0.0576i)37-s + ⋯
L(s)  = 1  + (0.568 − 0.822i)3-s + (−0.568 − 0.822i)4-s + (1.63 + 1.12i)7-s + (−0.354 − 0.935i)9-s − 12-s − 1.49·13-s + (−0.354 + 0.935i)16-s + (−0.136 − 1.12i)19-s + (1.85 − 0.704i)21-s + (−0.885 + 0.464i)25-s + (−0.970 − 0.239i)27-s − 1.98i·28-s + (1.32 + 1.17i)31-s + (−0.568 + 0.822i)36-s + (0.234 + 0.0576i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9842694443\)
\(L(\frac12)\) \(\approx\) \(0.9842694443\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.568 + 0.822i)T \)
157 \( 1 + (-0.568 + 0.822i)T \)
good2 \( 1 + (0.568 + 0.822i)T^{2} \)
5 \( 1 + (0.885 - 0.464i)T^{2} \)
7 \( 1 + (-1.63 - 1.12i)T + (0.354 + 0.935i)T^{2} \)
11 \( 1 + (-0.568 + 0.822i)T^{2} \)
13 \( 1 + 1.49T + T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 + (0.136 + 1.12i)T + (-0.970 + 0.239i)T^{2} \)
23 \( 1 + (-0.748 - 0.663i)T^{2} \)
29 \( 1 + (0.885 + 0.464i)T^{2} \)
31 \( 1 + (-1.32 - 1.17i)T + (0.120 + 0.992i)T^{2} \)
37 \( 1 + (-0.234 - 0.0576i)T + (0.885 + 0.464i)T^{2} \)
41 \( 1 + (-0.748 + 0.663i)T^{2} \)
43 \( 1 + (1.53 - 1.06i)T + (0.354 - 0.935i)T^{2} \)
47 \( 1 + (-0.885 + 0.464i)T^{2} \)
53 \( 1 + (-0.970 + 0.239i)T^{2} \)
59 \( 1 + (-0.970 - 0.239i)T^{2} \)
61 \( 1 + (-0.922 + 0.112i)T + (0.970 - 0.239i)T^{2} \)
67 \( 1 + (0.0854 + 0.704i)T + (-0.970 + 0.239i)T^{2} \)
71 \( 1 + (0.970 - 0.239i)T^{2} \)
73 \( 1 + (-0.393 - 0.271i)T + (0.354 + 0.935i)T^{2} \)
79 \( 1 + (0.764 + 1.45i)T + (-0.568 + 0.822i)T^{2} \)
83 \( 1 + (0.120 + 0.992i)T^{2} \)
89 \( 1 + (-0.120 - 0.992i)T^{2} \)
97 \( 1 + (0.114 + 0.464i)T + (-0.885 + 0.464i)T^{2} \)
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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

−11.35233205052696752038161504603, −10.01324497320551899572990698054, −9.120427311731015651901476422893, −8.446180807425484761369610011402, −7.63474876324205874974686717494, −6.45189403204830378580557644347, −5.26389216662725728489972026059, −4.66981743371648446597774422965, −2.62260786415784589466105544967, −1.61758927820961328996156005934, 2.27639377890431796298085942388, 3.83562840528565602642526227359, 4.45136918677368431487944362795, 5.23582402463196970977587309546, 7.27993418249049353773454990816, 7.995508944001491436893496017765, 8.406343431299639970493695366817, 9.789496796888436995549537083852, 10.21630088592330826401600832343, 11.42219874024643478375583477800

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