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Properties

Label 2-460-92.91-c1-0-18
Degree $2$
Conductor $460$
Sign $0.885 - 0.464i$
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  + (−1.39 + 0.234i)2-s + 1.37i·3-s + (1.89 − 0.653i)4-s − i·5-s + (−0.323 − 1.92i)6-s + 2.31·7-s + (−2.48 + 1.35i)8-s + 1.09·9-s + (0.234 + 1.39i)10-s + 1.65·11-s + (0.902 + 2.60i)12-s − 3.48·13-s + (−3.23 + 0.543i)14-s + 1.37·15-s + (3.14 − 2.47i)16-s − 5.56i·17-s + ⋯
L(s)  = 1  + (−0.986 + 0.165i)2-s + 0.796i·3-s + (0.945 − 0.326i)4-s − 0.447i·5-s + (−0.132 − 0.785i)6-s + 0.876·7-s + (−0.877 + 0.478i)8-s + 0.365·9-s + (0.0741 + 0.441i)10-s + 0.498·11-s + (0.260 + 0.752i)12-s − 0.965·13-s + (−0.864 + 0.145i)14-s + 0.356·15-s + (0.786 − 0.617i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 - 0.464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.885 - 0.464i$
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: \((2,\ 460,\ (\ :1/2),\ 0.885 - 0.464i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04117 + 0.256420i\)
\(L(\frac12)\) \(\approx\) \(1.04117 + 0.256420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.234i)T \)
5 \( 1 + iT \)
23 \( 1 + (0.716 + 4.74i)T \)
good3 \( 1 - 1.37iT - 3T^{2} \)
7 \( 1 - 2.31T + 7T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + 5.56iT - 17T^{2} \)
19 \( 1 - 6.63T + 19T^{2} \)
29 \( 1 - 4.94T + 29T^{2} \)
31 \( 1 - 3.53iT - 31T^{2} \)
37 \( 1 - 9.38iT - 37T^{2} \)
41 \( 1 + 6.37T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 - 8.60iT - 47T^{2} \)
53 \( 1 - 2.04iT - 53T^{2} \)
59 \( 1 + 9.50iT - 59T^{2} \)
61 \( 1 + 3.86iT - 61T^{2} \)
67 \( 1 - 4.67T + 67T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 6.51T + 73T^{2} \)
79 \( 1 + 4.23T + 79T^{2} \)
83 \( 1 + 5.44T + 83T^{2} \)
89 \( 1 + 7.48iT - 89T^{2} \)
97 \( 1 + 19.0iT - 97T^{2} \)
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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

−10.97165547420838589627547703540, −9.872890319005635695186103180201, −9.549653875838493773406011998084, −8.547762615540701585451468391919, −7.61221200308893236254331272873, −6.78398245660322353774942925452, −5.22957199904555680188182512413, −4.61766914126258357322487947689, −2.87071662653254084694873216495, −1.20763661591930341504996140256, 1.28074230515847051584430042060, 2.31108967176621629197059375264, 3.87390068518781088783110819845, 5.59192038607011900666171795179, 6.74859539679075791608546985919, 7.51978686645935939268049129268, 8.012589415733529956317704765418, 9.237765308362090843119586683582, 10.06992852417185506900852866928, 10.93316304251066116020996173867

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