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Properties

Label 2-471-157.12-c1-0-17
Degree $2$
Conductor $471$
Sign $0.710 - 0.704i$
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  + 2.06·2-s + (0.5 + 0.866i)3-s + 2.25·4-s + (1.20 + 2.07i)5-s + (1.03 + 1.78i)6-s + 1.31·7-s + 0.524·8-s + (−0.499 + 0.866i)9-s + (2.47 + 4.28i)10-s + (−1.50 − 2.60i)11-s + (1.12 + 1.95i)12-s + (−0.726 + 1.25i)13-s + 2.70·14-s + (−1.20 + 2.07i)15-s − 3.42·16-s + (−2.47 − 4.28i)17-s + ⋯
L(s)  = 1  + 1.45·2-s + (0.288 + 0.499i)3-s + 1.12·4-s + (0.536 + 0.929i)5-s + (0.421 + 0.729i)6-s + 0.496·7-s + 0.185·8-s + (−0.166 + 0.288i)9-s + (0.782 + 1.35i)10-s + (−0.453 − 0.786i)11-s + (0.325 + 0.563i)12-s + (−0.201 + 0.349i)13-s + 0.723·14-s + (−0.309 + 0.536i)15-s − 0.856·16-s + (−0.599 − 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.710 - 0.704i$
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: \((2,\ 471,\ (\ :1/2),\ 0.710 - 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.11217 + 1.28158i\)
\(L(\frac12)\) \(\approx\) \(3.11217 + 1.28158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.855 - 12.5i)T \)
good2 \( 1 - 2.06T + 2T^{2} \)
5 \( 1 + (-1.20 - 2.07i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.31T + 7T^{2} \)
11 \( 1 + (1.50 + 2.60i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.726 - 1.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.47 + 4.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.558 + 0.967i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.06T + 23T^{2} \)
29 \( 1 - 9.91T + 29T^{2} \)
31 \( 1 + (0.895 - 1.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.08 + 3.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.59T + 41T^{2} \)
43 \( 1 + (1.59 - 2.75i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.813 - 1.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.85 + 6.68i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.25T + 59T^{2} \)
61 \( 1 + (-3.20 - 5.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 + (-2.03 + 3.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.15 - 1.99i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3.95T + 79T^{2} \)
83 \( 1 + (1.00 - 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.254 + 0.441i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.07 - 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.18495709292080062928163873409, −10.58819632291364818674537534239, −9.434491281425404598899850592211, −8.454866645406508921097114019197, −7.05576926088380287945788970943, −6.29193448331254286654954588556, −5.18737498096872806187746989065, −4.49657277464590637269701109711, −3.12258572567454512239029107410, −2.52697338688060768881217521134, 1.68413893638848822801568984591, 2.89003953609356569741063295944, 4.39491736027109698668209723448, 5.01207016120550520809324385536, 5.97559389542069227864624317306, 6.94446926311065740803142276642, 8.168538134019605882431067135283, 8.950930940810657350323366811959, 10.12288253971853889489402842364, 11.24275178942245507530027964797

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