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Properties

Label 2-471-157.12-c1-0-24
Degree $2$
Conductor $471$
Sign $0.460 + 0.887i$
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  + 2.56·2-s + (−0.5 − 0.866i)3-s + 4.58·4-s + (−1.76 − 3.05i)5-s + (−1.28 − 2.22i)6-s − 0.0341·7-s + 6.64·8-s + (−0.499 + 0.866i)9-s + (−4.52 − 7.83i)10-s + (0.731 + 1.26i)11-s + (−2.29 − 3.97i)12-s + (0.617 − 1.07i)13-s − 0.0876·14-s + (−1.76 + 3.05i)15-s + 7.86·16-s + (1.48 + 2.56i)17-s + ⋯
L(s)  = 1  + 1.81·2-s + (−0.288 − 0.499i)3-s + 2.29·4-s + (−0.787 − 1.36i)5-s + (−0.523 − 0.907i)6-s − 0.0129·7-s + 2.34·8-s + (−0.166 + 0.288i)9-s + (−1.42 − 2.47i)10-s + (0.220 + 0.382i)11-s + (−0.662 − 1.14i)12-s + (0.171 − 0.296i)13-s − 0.0234·14-s + (−0.454 + 0.787i)15-s + 1.96·16-s + (0.359 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.460 + 0.887i$
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: \((2,\ 471,\ (\ :1/2),\ 0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79774 - 1.69984i\)
\(L(\frac12)\) \(\approx\) \(2.79774 - 1.69984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-11.6 - 4.53i)T \)
good2 \( 1 - 2.56T + 2T^{2} \)
5 \( 1 + (1.76 + 3.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.0341T + 7T^{2} \)
11 \( 1 + (-0.731 - 1.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.617 + 1.07i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.48 - 2.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.941 + 1.63i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.478T + 23T^{2} \)
29 \( 1 - 10.0T + 29T^{2} \)
31 \( 1 + (4.00 - 6.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.835 - 1.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.23T + 41T^{2} \)
43 \( 1 + (4.18 - 7.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.43 + 5.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.10 - 8.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.57T + 59T^{2} \)
61 \( 1 + (-6.93 - 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 - 5.26T + 67T^{2} \)
71 \( 1 + (-4.71 + 8.16i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.08 + 14.0i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.05T + 79T^{2} \)
83 \( 1 + (-0.106 + 0.185i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.20 + 15.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.60 + 9.70i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.42856069557484475807476172420, −10.42015975556000862191705472042, −8.797814914500059349855336117605, −7.87832933702072947972798314563, −6.87604870721389944412093414321, −5.90777289926705994042514340677, −4.91115489926285638271156576626, −4.35746002093791859040313322548, −3.12577097259526182170831204619, −1.42245574808721451973741920267, 2.61393824620613290899073683082, 3.52269163068594279689134162042, 4.21914373237299215930917331457, 5.38352917576005544509642705954, 6.40636816395319405558965684988, 6.97578089256172328195507444964, 8.090557769246591716271929440583, 9.811469503539518463881102395791, 10.87086330529122682326006991993, 11.30943317795424044727271184991

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