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Properties

Label 2-460-92.91-c1-0-43
Degree $2$
Conductor $460$
Sign $-0.574 - 0.818i$
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  + (−0.151 − 1.40i)2-s − 2.99i·3-s + (−1.95 + 0.425i)4-s − i·5-s + (−4.21 + 0.453i)6-s + 3.98·7-s + (0.893 + 2.68i)8-s − 5.98·9-s + (−1.40 + 0.151i)10-s − 5.79·11-s + (1.27 + 5.85i)12-s − 5.59·13-s + (−0.602 − 5.59i)14-s − 2.99·15-s + (3.63 − 1.66i)16-s − 1.88i·17-s + ⋯
L(s)  = 1  + (−0.106 − 0.994i)2-s − 1.73i·3-s + (−0.977 + 0.212i)4-s − 0.447i·5-s + (−1.72 + 0.185i)6-s + 1.50·7-s + (0.315 + 0.948i)8-s − 1.99·9-s + (−0.444 + 0.0478i)10-s − 1.74·11-s + (0.368 + 1.69i)12-s − 1.55·13-s + (−0.160 − 1.49i)14-s − 0.774·15-s + (0.909 − 0.415i)16-s − 0.456i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.574 - 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.574 - 0.818i$
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: \((2,\ 460,\ (\ :1/2),\ -0.574 - 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.441970 + 0.850728i\)
\(L(\frac12)\) \(\approx\) \(0.441970 + 0.850728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.151 + 1.40i)T \)
5 \( 1 + iT \)
23 \( 1 + (-4.42 + 1.85i)T \)
good3 \( 1 + 2.99iT - 3T^{2} \)
7 \( 1 - 3.98T + 7T^{2} \)
11 \( 1 + 5.79T + 11T^{2} \)
13 \( 1 + 5.59T + 13T^{2} \)
17 \( 1 + 1.88iT - 17T^{2} \)
19 \( 1 - 3.03T + 19T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 + 4.43iT - 31T^{2} \)
37 \( 1 - 4.01iT - 37T^{2} \)
41 \( 1 + 2.68T + 41T^{2} \)
43 \( 1 - 4.78T + 43T^{2} \)
47 \( 1 + 6.45iT - 47T^{2} \)
53 \( 1 + 8.62iT - 53T^{2} \)
59 \( 1 + 1.55iT - 59T^{2} \)
61 \( 1 + 6.06iT - 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 + 5.26iT - 71T^{2} \)
73 \( 1 - 2.20T + 73T^{2} \)
79 \( 1 - 13.1T + 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 - 8.41iT - 89T^{2} \)
97 \( 1 - 2.87iT - 97T^{2} \)
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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

−10.78954446804749818840699349094, −9.621551956258333915691909472452, −8.311876130051242789514409996929, −7.929457395327154634284168809856, −7.19768456071297107840759877993, −5.25483354048829077739661550086, −4.96886432780788319119972028431, −2.70009590510981186090186997322, −1.99565578732841433138515620392, −0.61214061972460443618340604014, 2.85512985426219632672877783302, 4.35438417574474404048069356418, 5.10999497199785456639401690401, 5.47510199844017187060771949450, 7.34462840742329801187190302124, 7.971609101245998314781228495460, 8.941614086313297277975269415817, 9.884112496296329222903266765096, 10.52997501818797086249122821872, 11.15746801067043255971326232320

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