Properties

Label 2-460-460.167-c1-0-56
Degree $2$
Conductor $460$
Sign $0.773 + 0.633i$
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  + (1.40 + 0.148i)2-s + (0.0994 + 0.182i)3-s + (1.95 + 0.418i)4-s + (0.0408 − 2.23i)5-s + (0.112 + 0.271i)6-s + (−2.09 − 2.80i)7-s + (2.68 + 0.880i)8-s + (1.59 − 2.48i)9-s + (0.390 − 3.13i)10-s + (−0.956 + 0.436i)11-s + (0.118 + 0.398i)12-s + (1.28 + 0.959i)13-s + (−2.53 − 4.25i)14-s + (0.411 − 0.214i)15-s + (3.64 + 1.63i)16-s + (−3.78 + 0.270i)17-s + ⋯
L(s)  = 1  + (0.994 + 0.105i)2-s + (0.0574 + 0.105i)3-s + (0.977 + 0.209i)4-s + (0.0182 − 0.999i)5-s + (0.0460 + 0.110i)6-s + (−0.793 − 1.06i)7-s + (0.950 + 0.311i)8-s + (0.532 − 0.829i)9-s + (0.123 − 0.992i)10-s + (−0.288 + 0.131i)11-s + (0.0341 + 0.114i)12-s + (0.355 + 0.266i)13-s + (−0.677 − 1.13i)14-s + (0.106 − 0.0555i)15-s + (0.912 + 0.409i)16-s + (−0.917 + 0.0656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37882 - 0.849731i\)
\(L(\frac12)\) \(\approx\) \(2.37882 - 0.849731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.148i)T \)
5 \( 1 + (-0.0408 + 2.23i)T \)
23 \( 1 + (-2.44 - 4.12i)T \)
good3 \( 1 + (-0.0994 - 0.182i)T + (-1.62 + 2.52i)T^{2} \)
7 \( 1 + (2.09 + 2.80i)T + (-1.97 + 6.71i)T^{2} \)
11 \( 1 + (0.956 - 0.436i)T + (7.20 - 8.31i)T^{2} \)
13 \( 1 + (-1.28 - 0.959i)T + (3.66 + 12.4i)T^{2} \)
17 \( 1 + (3.78 - 0.270i)T + (16.8 - 2.41i)T^{2} \)
19 \( 1 + (-1.55 - 1.79i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-4.94 - 4.28i)T + (4.12 + 28.7i)T^{2} \)
31 \( 1 + (-0.00306 - 0.0104i)T + (-26.0 + 16.7i)T^{2} \)
37 \( 1 + (-2.06 - 9.49i)T + (-33.6 + 15.3i)T^{2} \)
41 \( 1 + (-3.82 + 2.45i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.54 + 4.66i)T + (-23.2 + 36.1i)T^{2} \)
47 \( 1 + (6.89 - 6.89i)T - 47iT^{2} \)
53 \( 1 + (-0.984 - 1.31i)T + (-14.9 + 50.8i)T^{2} \)
59 \( 1 + (0.811 + 5.64i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-0.228 + 0.0671i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (5.38 + 14.4i)T + (-50.6 + 43.8i)T^{2} \)
71 \( 1 + (-0.804 - 0.367i)T + (46.4 + 53.6i)T^{2} \)
73 \( 1 + (-7.19 - 0.514i)T + (72.2 + 10.3i)T^{2} \)
79 \( 1 + (-1.38 - 9.66i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (0.867 + 3.98i)T + (-75.4 + 34.4i)T^{2} \)
89 \( 1 + (1.49 - 5.10i)T + (-74.8 - 48.1i)T^{2} \)
97 \( 1 + (14.8 + 3.23i)T + (88.2 + 40.2i)T^{2} \)
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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

−11.10496518993858662853979867322, −10.10047261489940328164445049231, −9.295579004605908216120195777392, −8.065759661840338364469886859016, −6.97334812185332424722481782540, −6.32191607215406089956855463163, −5.00292408605892936714966445953, −4.14685045971370691173989553741, −3.29480228246526443955377402501, −1.31694749327550995679884726556, 2.33332918146672381061600814935, 2.93448199695928719922982625304, 4.33141420398397755980380586988, 5.53887681444443622316970580121, 6.41009520396653279310220852075, 7.13116975464577532434983266023, 8.260917695875488457717106811226, 9.615724791765294693516400456033, 10.55046746356373404618621867743, 11.17402433206456505820014040865

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