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Properties

Label 2-459-17.16-c1-0-7
Degree $2$
Conductor $459$
Sign $0.0649 - 0.997i$
Analytic cond. $3.66513$
Root an. cond. $1.91445$
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  + 1.50·2-s + 0.267·4-s + 1.73i·5-s + 4.11i·7-s − 2.60·8-s + 2.60i·10-s + 2i·11-s − 4.46·13-s + 6.19i·14-s − 4.46·16-s + (4.11 + 0.267i)17-s + 7.19·19-s + 0.464i·20-s + 3.01i·22-s − 8.46i·23-s + ⋯
L(s)  = 1  + 1.06·2-s + 0.133·4-s + 0.774i·5-s + 1.55i·7-s − 0.922·8-s + 0.824i·10-s + 0.603i·11-s − 1.23·13-s + 1.65i·14-s − 1.11·16-s + (0.997 + 0.0649i)17-s + 1.65·19-s + 0.103i·20-s + 0.642i·22-s − 1.76i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0649 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $0.0649 - 0.997i$
Analytic conductor: \(3.66513\)
Root analytic conductor: \(1.91445\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 459,\ (\ :1/2),\ 0.0649 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40497 + 1.31645i\)
\(L(\frac12)\) \(\approx\) \(1.40497 + 1.31645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (-4.11 - 0.267i)T \)
good2 \( 1 - 1.50T + 2T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 - 4.11iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + 4.46T + 13T^{2} \)
19 \( 1 - 7.19T + 19T^{2} \)
23 \( 1 + 8.46iT - 23T^{2} \)
29 \( 1 + 1.73iT - 29T^{2} \)
31 \( 1 - 6.02iT - 31T^{2} \)
37 \( 1 - 9.33iT - 37T^{2} \)
41 \( 1 + 6.66iT - 41T^{2} \)
43 \( 1 - 0.267T + 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 + 1.90T + 53T^{2} \)
59 \( 1 + 7.12T + 59T^{2} \)
61 \( 1 - 1.10iT - 61T^{2} \)
67 \( 1 - 4.26T + 67T^{2} \)
71 \( 1 + 5.53iT - 71T^{2} \)
73 \( 1 - 7.12iT - 73T^{2} \)
79 \( 1 + 10.1iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 2.20T + 89T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−11.83852180684890048571261193798, −10.36937513019852011304626270656, −9.536103849600048781692236324721, −8.653403562478189255939229047592, −7.38180990726082442508316689324, −6.37822204744743383377617210464, −5.38583834103476527280795697682, −4.76015807797975356260386837211, −3.15244222341905479560143880175, −2.52889466647988725815317148312, 0.898795883062518833635365205675, 3.17927000887785320352562838032, 4.03527470398510499063589499791, 5.05816079572590758511993994614, 5.68982671826630413061713561198, 7.22940536220388110355430488666, 7.80492582853879600134751797548, 9.316793467423443972990311055414, 9.796995989489913877562029231619, 11.09575158672502522324963815660

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