Properties

Label 2-471-157.144-c1-0-22
Degree $2$
Conductor $471$
Sign $0.996 + 0.0846i$
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  + 2.49·2-s + (−0.5 + 0.866i)3-s + 4.23·4-s + (1.51 − 2.62i)5-s + (−1.24 + 2.16i)6-s − 1.77·7-s + 5.58·8-s + (−0.499 − 0.866i)9-s + (3.78 − 6.56i)10-s + (0.957 − 1.65i)11-s + (−2.11 + 3.66i)12-s + (1.94 + 3.36i)13-s − 4.42·14-s + (1.51 + 2.62i)15-s + 5.47·16-s + (−2.89 + 5.01i)17-s + ⋯
L(s)  = 1  + 1.76·2-s + (−0.288 + 0.499i)3-s + 2.11·4-s + (0.678 − 1.17i)5-s + (−0.509 + 0.882i)6-s − 0.670·7-s + 1.97·8-s + (−0.166 − 0.288i)9-s + (1.19 − 2.07i)10-s + (0.288 − 0.499i)11-s + (−0.611 + 1.05i)12-s + (0.539 + 0.934i)13-s − 1.18·14-s + (0.391 + 0.678i)15-s + 1.36·16-s + (−0.702 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.996 + 0.0846i$
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: \((2,\ 471,\ (\ :1/2),\ 0.996 + 0.0846i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.58793 - 0.152133i\)
\(L(\frac12)\) \(\approx\) \(3.58793 - 0.152133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8.67 - 9.04i)T \)
good2 \( 1 - 2.49T + 2T^{2} \)
5 \( 1 + (-1.51 + 2.62i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.77T + 7T^{2} \)
11 \( 1 + (-0.957 + 1.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.94 - 3.36i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.89 - 5.01i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.60 + 2.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 8.14T + 23T^{2} \)
29 \( 1 + 1.03T + 29T^{2} \)
31 \( 1 + (-2.29 - 3.98i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.66 - 8.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.436T + 41T^{2} \)
43 \( 1 + (0.232 + 0.402i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.04 + 8.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.137 - 0.238i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.74T + 59T^{2} \)
61 \( 1 + (-5.13 + 8.89i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 6.32T + 67T^{2} \)
71 \( 1 + (7.15 + 12.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.85 + 3.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.892T + 79T^{2} \)
83 \( 1 + (-7.66 - 13.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.61 + 6.25i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.18 + 5.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.42888268541645167831049964544, −10.30904643442088102924355766065, −9.313625255492238309395070702898, −8.392143127349552564153959133191, −6.49404816212826167046124782511, −6.19300756398121151961346547773, −5.14989445851620813006609102831, −4.32884444095468955510803104408, −3.48052552656864564509588384541, −1.84538229636468343907640262188, 2.21541613610161567372879525137, 3.04535843277054552347745893738, 4.18288319456138862212761135890, 5.62698702685278758054471762212, 6.14300229542363975821021548253, 6.86453171051723998495122213401, 7.72379482533976444655750905163, 9.624160558998106269968833778122, 10.45958122530864008115652328799, 11.37041037679551394099577298053

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