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Properties

Label 2-471-157.144-c1-0-25
Degree $2$
Conductor $471$
Sign $0.909 + 0.414i$
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  + 2.74·2-s + (0.5 − 0.866i)3-s + 5.55·4-s + (0.277 − 0.481i)5-s + (1.37 − 2.38i)6-s − 4.42·7-s + 9.76·8-s + (−0.499 − 0.866i)9-s + (0.763 − 1.32i)10-s + (−1.93 + 3.35i)11-s + (2.77 − 4.80i)12-s + (0.286 + 0.495i)13-s − 12.1·14-s + (−0.277 − 0.481i)15-s + 15.7·16-s + (2.88 − 4.99i)17-s + ⋯
L(s)  = 1  + 1.94·2-s + (0.288 − 0.499i)3-s + 2.77·4-s + (0.124 − 0.215i)5-s + (0.561 − 0.971i)6-s − 1.67·7-s + 3.45·8-s + (−0.166 − 0.288i)9-s + (0.241 − 0.418i)10-s + (−0.584 + 1.01i)11-s + (0.801 − 1.38i)12-s + (0.0793 + 0.137i)13-s − 3.25·14-s + (−0.0717 − 0.124i)15-s + 3.93·16-s + (0.699 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.909 + 0.414i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.909 + 0.414i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.12457 - 0.895609i\)
\(L(\frac12)\) \(\approx\) \(4.12457 - 0.895609i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.13 + 11.4i)T \)
good2 \( 1 - 2.74T + 2T^{2} \)
5 \( 1 + (-0.277 + 0.481i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 + (1.93 - 3.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.286 - 0.495i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.88 + 4.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.35 - 4.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.19T + 23T^{2} \)
29 \( 1 + 10.3T + 29T^{2} \)
31 \( 1 + (1.89 + 3.27i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 2.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 + (5.04 + 8.74i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.70 - 2.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.565 - 0.978i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.90T + 59T^{2} \)
61 \( 1 + (0.175 - 0.303i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 2.06T + 67T^{2} \)
71 \( 1 + (-4.68 - 8.10i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.74 + 11.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 + (3.41 + 5.91i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.76 + 4.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.40 + 14.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.40898177748680940609641247971, −10.22192723957398691404996436979, −9.412858619611980943001825682095, −7.55146128589036094524855400319, −7.08537808938808891611755409048, −6.05379415445372497190065551489, −5.34660637793782246195409174937, −3.99775944501173351742606770906, −3.13279739015160618441337949151, −2.08773779382322451236568336864, 2.54266232063189589938458866075, 3.34735436324404529048408734806, 4.03632347700165367371129050426, 5.46601482984469671502514373470, 6.08262044185083471569930416962, 6.89446240620892901484559383776, 8.132013150305021649662004204733, 9.584193543898005515197587730583, 10.63175471190694140483055630921, 11.04205144477914575688197746913

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