Properties

Label 2-471-471.14-c0-0-0
Degree $2$
Conductor $471$
Sign $0.813 - 0.581i$
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  + (−0.354 − 0.935i)3-s + (−0.354 + 0.935i)4-s + (0.688 + 1.81i)7-s + (−0.748 + 0.663i)9-s + 12-s + 0.241·13-s + (−0.748 − 0.663i)16-s + (0.688 − 0.169i)19-s + (1.45 − 1.28i)21-s + (0.568 − 0.822i)25-s + (0.885 + 0.464i)27-s − 1.94·28-s + (0.136 + 1.12i)31-s + (−0.354 − 0.935i)36-s + (−1.71 − 0.902i)37-s + ⋯
L(s)  = 1  + (−0.354 − 0.935i)3-s + (−0.354 + 0.935i)4-s + (0.688 + 1.81i)7-s + (−0.748 + 0.663i)9-s + 12-s + 0.241·13-s + (−0.748 − 0.663i)16-s + (0.688 − 0.169i)19-s + (1.45 − 1.28i)21-s + (0.568 − 0.822i)25-s + (0.885 + 0.464i)27-s − 1.94·28-s + (0.136 + 1.12i)31-s + (−0.354 − 0.935i)36-s + (−1.71 − 0.902i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(0.235059\)
Root analytic conductor: \(0.484829\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :0),\ 0.813 - 0.581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7466718784\)
\(L(\frac12)\) \(\approx\) \(0.7466718784\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.354 + 0.935i)T \)
157 \( 1 + (0.354 + 0.935i)T \)
good2 \( 1 + (0.354 - 0.935i)T^{2} \)
5 \( 1 + (-0.568 + 0.822i)T^{2} \)
7 \( 1 + (-0.688 - 1.81i)T + (-0.748 + 0.663i)T^{2} \)
11 \( 1 + (0.354 + 0.935i)T^{2} \)
13 \( 1 - 0.241T + T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + (-0.688 + 0.169i)T + (0.885 - 0.464i)T^{2} \)
23 \( 1 + (-0.120 - 0.992i)T^{2} \)
29 \( 1 + (-0.568 - 0.822i)T^{2} \)
31 \( 1 + (-0.136 - 1.12i)T + (-0.970 + 0.239i)T^{2} \)
37 \( 1 + (1.71 + 0.902i)T + (0.568 + 0.822i)T^{2} \)
41 \( 1 + (-0.120 + 0.992i)T^{2} \)
43 \( 1 + (-0.530 + 1.39i)T + (-0.748 - 0.663i)T^{2} \)
47 \( 1 + (-0.568 + 0.822i)T^{2} \)
53 \( 1 + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (-0.885 - 0.464i)T^{2} \)
61 \( 1 + (1.10 - 0.271i)T + (0.885 - 0.464i)T^{2} \)
67 \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \)
71 \( 1 + (-0.885 + 0.464i)T^{2} \)
73 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
79 \( 1 + (0.402 - 0.583i)T + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + (0.970 - 0.239i)T^{2} \)
89 \( 1 + (0.970 - 0.239i)T^{2} \)
97 \( 1 + (-1.56 + 0.822i)T + (0.568 - 0.822i)T^{2} \)
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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

−11.73314584312763694221201408901, −10.72226748286046332327578004753, −9.020384734311007429835970846306, −8.643187898974223739536314360002, −7.79577330777980739183557071567, −6.80597061588173867907849122299, −5.63230892463637508235188073176, −4.88889500623154806050680027607, −3.10915271065040703393501263450, −2.04796592169459098043352833249, 1.16673175893700056709446876837, 3.58486516713060900435747031011, 4.51417395741628469178950658269, 5.21095813101250995910446385336, 6.37092088947012222282156281639, 7.45677460843016308980612125417, 8.638295474889987195446033291008, 9.724361485997275184318352875990, 10.23786177750731710031482070645, 11.01240711636923100664948386504

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