Properties

Label 2-471-157.12-c1-0-2
Degree $2$
Conductor $471$
Sign $-0.682 - 0.730i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.10·2-s + (−0.5 − 0.866i)3-s − 0.772·4-s + (1.74 + 3.02i)5-s + (−0.553 − 0.959i)6-s − 4.63·7-s − 3.07·8-s + (−0.499 + 0.866i)9-s + (1.93 + 3.35i)10-s + (−0.619 − 1.07i)11-s + (0.386 + 0.669i)12-s + (−3.04 + 5.26i)13-s − 5.13·14-s + (1.74 − 3.02i)15-s − 1.85·16-s + (1.45 + 2.52i)17-s + ⋯
L(s)  = 1  + 0.783·2-s + (−0.288 − 0.499i)3-s − 0.386·4-s + (0.781 + 1.35i)5-s + (−0.226 − 0.391i)6-s − 1.75·7-s − 1.08·8-s + (−0.166 + 0.288i)9-s + (0.612 + 1.06i)10-s + (−0.186 − 0.323i)11-s + (0.111 + 0.193i)12-s + (−0.843 + 1.46i)13-s − 1.37·14-s + (0.451 − 0.781i)15-s − 0.464·16-s + (0.353 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.682 - 0.730i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ -0.682 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290817 + 0.670067i\)
\(L(\frac12)\) \(\approx\) \(0.290817 + 0.670067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
157 \( 1 + (12.4 + 1.22i)T \)
good2 \( 1 - 1.10T + 2T^{2} \)
5 \( 1 + (-1.74 - 3.02i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.63T + 7T^{2} \)
11 \( 1 + (0.619 + 1.07i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.04 - 5.26i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.45 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.53 + 6.11i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.72T + 23T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + (2.01 - 3.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.339 + 0.588i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.45T + 41T^{2} \)
43 \( 1 + (2.77 - 4.79i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.11 - 7.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.58 - 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.51T + 59T^{2} \)
61 \( 1 + (-0.973 - 1.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + (5.39 - 9.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.48 + 9.49i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 5.38T + 79T^{2} \)
83 \( 1 + (-6.55 + 11.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.26 + 16.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.36 - 5.82i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.44524095488122869275228539848, −10.47606758139273541470336558748, −9.594885571412798660150342797882, −8.959625186907743907812102600474, −7.11825438425999452508671095078, −6.49494502281392167472258746907, −6.05205357326855642977827250779, −4.67394448131316227497132184846, −3.25555168960681642471312167323, −2.55203142054720935528379896266, 0.34042579715424217172848884217, 2.86184358028181933831738530485, 3.94339813541112171421039998060, 5.15754878887462330047700168125, 5.54589980946196825848729960919, 6.51824432981606085966548763385, 8.168777025954930141202978677773, 9.232519730391178522705238115529, 9.769622314583008966346511633220, 10.30622161737214093823787977195

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