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Properties

Label 2-460-92.91-c1-0-46
Degree $2$
Conductor $460$
Sign $-0.998 - 0.0501i$
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  + (0.949 − 1.04i)2-s − 2.47i·3-s + (−0.198 − 1.99i)4-s + i·5-s + (−2.59 − 2.35i)6-s − 1.26·7-s + (−2.27 − 1.68i)8-s − 3.14·9-s + (1.04 + 0.949i)10-s − 3.32·11-s + (−4.93 + 0.491i)12-s + 1.65·13-s + (−1.19 + 1.32i)14-s + 2.47·15-s + (−3.92 + 0.788i)16-s − 6.70i·17-s + ⋯
L(s)  = 1  + (0.671 − 0.741i)2-s − 1.43i·3-s + (−0.0990 − 0.995i)4-s + 0.447i·5-s + (−1.06 − 0.960i)6-s − 0.476·7-s + (−0.804 − 0.594i)8-s − 1.04·9-s + (0.331 + 0.300i)10-s − 1.00·11-s + (−1.42 + 0.141i)12-s + 0.457·13-s + (−0.319 + 0.353i)14-s + 0.640·15-s + (−0.980 + 0.197i)16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.998 - 0.0501i$
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: \((2,\ 460,\ (\ :1/2),\ -0.998 - 0.0501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0402168 + 1.60408i\)
\(L(\frac12)\) \(\approx\) \(0.0402168 + 1.60408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.949 + 1.04i)T \)
5 \( 1 - iT \)
23 \( 1 + (-4.74 - 0.713i)T \)
good3 \( 1 + 2.47iT - 3T^{2} \)
7 \( 1 + 1.26T + 7T^{2} \)
11 \( 1 + 3.32T + 11T^{2} \)
13 \( 1 - 1.65T + 13T^{2} \)
17 \( 1 + 6.70iT - 17T^{2} \)
19 \( 1 - 0.390T + 19T^{2} \)
29 \( 1 - 8.11T + 29T^{2} \)
31 \( 1 - 2.86iT - 31T^{2} \)
37 \( 1 + 6.44iT - 37T^{2} \)
41 \( 1 - 4.47T + 41T^{2} \)
43 \( 1 - 2.71T + 43T^{2} \)
47 \( 1 + 0.827iT - 47T^{2} \)
53 \( 1 - 8.17iT - 53T^{2} \)
59 \( 1 + 4.19iT - 59T^{2} \)
61 \( 1 - 5.01iT - 61T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 + 9.80iT - 71T^{2} \)
73 \( 1 - 3.36T + 73T^{2} \)
79 \( 1 + 15.5T + 79T^{2} \)
83 \( 1 + 1.10T + 83T^{2} \)
89 \( 1 + 8.50iT - 89T^{2} \)
97 \( 1 - 7.78iT - 97T^{2} \)
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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

−10.89642933411971261079761409316, −9.934089056337657662840899734496, −8.859594010884040600918056623765, −7.53623309337748080320005579247, −6.83013770794132854880909394267, −5.93821688258015224378837232908, −4.84972125218333879403890842508, −3.13180311884212406410512548372, −2.41902420656386910629252735549, −0.812653423846397003873819988608, 2.95994986699415783684662132998, 3.95987218741784708451345787304, 4.79628161776253688491776851730, 5.63617652927001928928443763920, 6.61988642972183100519055726238, 8.085904703824226738722764538935, 8.665933699839563439888464041008, 9.722157683595433054703220782669, 10.53436483714006735683751346336, 11.42525310653230198788683350785

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