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Properties

Label 2-471-157.12-c1-0-26
Degree $2$
Conductor $471$
Sign $-0.741 + 0.671i$
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  + 1.32·2-s + (0.5 + 0.866i)3-s − 0.245·4-s + (−2.05 − 3.55i)5-s + (0.662 + 1.14i)6-s − 3.69·7-s − 2.97·8-s + (−0.499 + 0.866i)9-s + (−2.71 − 4.71i)10-s + (0.214 + 0.371i)11-s + (−0.122 − 0.212i)12-s + (1.17 − 2.03i)13-s − 4.89·14-s + (2.05 − 3.55i)15-s − 3.44·16-s + (0.765 + 1.32i)17-s + ⋯
L(s)  = 1  + 0.936·2-s + (0.288 + 0.499i)3-s − 0.122·4-s + (−0.918 − 1.59i)5-s + (0.270 + 0.468i)6-s − 1.39·7-s − 1.05·8-s + (−0.166 + 0.288i)9-s + (−0.859 − 1.48i)10-s + (0.0647 + 0.112i)11-s + (−0.0354 − 0.0614i)12-s + (0.326 − 0.565i)13-s − 1.30·14-s + (0.530 − 0.918i)15-s − 0.862·16-s + (0.185 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $-0.741 + 0.671i$
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: \((2,\ 471,\ (\ :1/2),\ -0.741 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.274481 - 0.712145i\)
\(L(\frac12)\) \(\approx\) \(0.274481 - 0.712145i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 + 12.4i)T \)
good2 \( 1 - 1.32T + 2T^{2} \)
5 \( 1 + (2.05 + 3.55i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.69T + 7T^{2} \)
11 \( 1 + (-0.214 - 0.371i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.17 + 2.03i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.765 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.35 + 5.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.95T + 23T^{2} \)
29 \( 1 - 6.08T + 29T^{2} \)
31 \( 1 + (2.68 - 4.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.23 + 7.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.93T + 41T^{2} \)
43 \( 1 + (-4.73 + 8.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.29 - 5.71i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.67 + 8.09i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 + (4.87 + 8.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 13.8T + 67T^{2} \)
71 \( 1 + (-0.536 + 0.929i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.85 - 4.94i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 7.18T + 79T^{2} \)
83 \( 1 + (5.58 - 9.68i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.330 - 0.572i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.13 + 1.96i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.75241681819003043864615240504, −9.495828233561386694570403426095, −8.933323609719900634069599374898, −8.264905198059487914680435718984, −6.78336987056804644659082390905, −5.55746721698933933843950543070, −4.72934151954641026841617560306, −3.92758419150715157141587049809, −3.08858334637976567047207306610, −0.32528831131428822860686810638, 2.80147817593918892837119512835, 3.39971459950271567504698791453, 4.25261947806224326494820026879, 6.19652975891274987674637618557, 6.41631092760406598072605402211, 7.46973119013654617135751450753, 8.545998378290338020044999969410, 9.694779596924722590709660545315, 10.54910379344334818871919858944, 11.75044899727587449528938807925

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