Properties

Label 2-460-92.91-c1-0-21
Degree $2$
Conductor $460$
Sign $0.155 - 0.987i$
Analytic cond. $3.67311$
Root an. cond. $1.91653$
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  + (0.637 + 1.26i)2-s + 0.348i·3-s + (−1.18 + 1.61i)4-s i·5-s + (−0.439 + 0.222i)6-s + 4.89·7-s + (−2.78 − 0.469i)8-s + 2.87·9-s + (1.26 − 0.637i)10-s + 2.68·11-s + (−0.560 − 0.412i)12-s − 3.09·13-s + (3.12 + 6.17i)14-s + 0.348·15-s + (−1.18 − 3.82i)16-s − 0.769i·17-s + ⋯
L(s)  = 1  + (0.451 + 0.892i)2-s + 0.200i·3-s + (−0.593 + 0.805i)4-s − 0.447i·5-s + (−0.179 + 0.0906i)6-s + 1.84·7-s + (−0.986 − 0.166i)8-s + 0.959·9-s + (0.399 − 0.201i)10-s + 0.808·11-s + (−0.161 − 0.119i)12-s − 0.859·13-s + (0.833 + 1.65i)14-s + 0.0898·15-s + (−0.296 − 0.955i)16-s − 0.186i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.155 - 0.987i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52804 + 1.30650i\)
\(L(\frac12)\) \(\approx\) \(1.52804 + 1.30650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.637 - 1.26i)T \)
5 \( 1 + iT \)
23 \( 1 + (-2.20 - 4.25i)T \)
good3 \( 1 - 0.348iT - 3T^{2} \)
7 \( 1 - 4.89T + 7T^{2} \)
11 \( 1 - 2.68T + 11T^{2} \)
13 \( 1 + 3.09T + 13T^{2} \)
17 \( 1 + 0.769iT - 17T^{2} \)
19 \( 1 + 4.89T + 19T^{2} \)
29 \( 1 + 1.72T + 29T^{2} \)
31 \( 1 - 8.95iT - 31T^{2} \)
37 \( 1 + 2.12iT - 37T^{2} \)
41 \( 1 + 5.04T + 41T^{2} \)
43 \( 1 + 0.739T + 43T^{2} \)
47 \( 1 + 9.51iT - 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 - 1.87iT - 59T^{2} \)
61 \( 1 + 8.05iT - 61T^{2} \)
67 \( 1 + 5.76T + 67T^{2} \)
71 \( 1 + 0.619iT - 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 6.71T + 79T^{2} \)
83 \( 1 + 4.75T + 83T^{2} \)
89 \( 1 + 9.87iT - 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−11.51090384366219705179888552959, −10.30982742230764582411854599918, −9.134047134554002592535829098086, −8.441987013480798003043887223229, −7.51430657939360435661328350609, −6.77398377635917619382630412435, −5.23093924336224843248263583134, −4.77279195487591729979947592018, −3.84126815197696197664909941604, −1.72019675183603140335098774017, 1.43662845010846257209249587670, 2.40846990631220874342852280966, 4.22853652583289068989672085068, 4.62616135851276057956409246167, 5.99790603381046790339259821824, 7.17539770137230177241035934301, 8.198380358171918505675126292549, 9.226266597885082571290549669200, 10.28894232006759537283753112948, 10.94937164092418779441325377768

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