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Properties

Label 2-460-115.22-c1-0-8
Degree $2$
Conductor $460$
Sign $0.474 + 0.880i$
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  + (0.237 − 0.237i)3-s + (−2.23 + 0.123i)5-s + (1.69 − 1.69i)7-s + 2.88i·9-s − 5.55i·11-s + (2.65 − 2.65i)13-s + (−0.500 + 0.558i)15-s + (0.435 − 0.435i)17-s + 1.92·19-s − 0.803i·21-s + (0.546 − 4.76i)23-s + (4.96 − 0.551i)25-s + (1.39 + 1.39i)27-s − 7.41i·29-s − 0.525·31-s + ⋯
L(s)  = 1  + (0.136 − 0.136i)3-s + (−0.998 + 0.0552i)5-s + (0.640 − 0.640i)7-s + 0.962i·9-s − 1.67i·11-s + (0.736 − 0.736i)13-s + (−0.129 + 0.144i)15-s + (0.105 − 0.105i)17-s + 0.440·19-s − 0.175i·21-s + (0.113 − 0.993i)23-s + (0.993 − 0.110i)25-s + (0.268 + 0.268i)27-s − 1.37i·29-s − 0.0944·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10234 - 0.657879i\)
\(L(\frac12)\) \(\approx\) \(1.10234 - 0.657879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.23 - 0.123i)T \)
23 \( 1 + (-0.546 + 4.76i)T \)
good3 \( 1 + (-0.237 + 0.237i)T - 3iT^{2} \)
7 \( 1 + (-1.69 + 1.69i)T - 7iT^{2} \)
11 \( 1 + 5.55iT - 11T^{2} \)
13 \( 1 + (-2.65 + 2.65i)T - 13iT^{2} \)
17 \( 1 + (-0.435 + 0.435i)T - 17iT^{2} \)
19 \( 1 - 1.92T + 19T^{2} \)
29 \( 1 + 7.41iT - 29T^{2} \)
31 \( 1 + 0.525T + 31T^{2} \)
37 \( 1 + (1.94 - 1.94i)T - 37iT^{2} \)
41 \( 1 + 6.78T + 41T^{2} \)
43 \( 1 + (-1.88 - 1.88i)T + 43iT^{2} \)
47 \( 1 + (-7.54 - 7.54i)T + 47iT^{2} \)
53 \( 1 + (0.467 + 0.467i)T + 53iT^{2} \)
59 \( 1 + 7.72iT - 59T^{2} \)
61 \( 1 - 0.714iT - 61T^{2} \)
67 \( 1 + (4.97 - 4.97i)T - 67iT^{2} \)
71 \( 1 + 4.93T + 71T^{2} \)
73 \( 1 + (-0.547 + 0.547i)T - 73iT^{2} \)
79 \( 1 - 9.07T + 79T^{2} \)
83 \( 1 + (-10.7 - 10.7i)T + 83iT^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (3.99 - 3.99i)T - 97iT^{2} \)
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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

−11.00824056677528892495441190547, −10.36079820483428975810759445925, −8.720301856948762638724102045716, −8.095108370540922264712540911954, −7.57640067898290363439964006876, −6.23686263544956114435263415086, −5.07596976025471472562977732344, −3.97041706392840127473517764377, −2.90482275779308714683114571212, −0.873735635301177111644858106355, 1.66855530778168813375954108466, 3.40616595155027579764931898190, 4.34050035435803039872346803339, 5.36340435117106393588017723295, 6.82290272981612130343132262103, 7.47326265115328562392440580496, 8.680924076056669222350810057314, 9.192585958888451020316772374880, 10.32077165952980082395542304518, 11.45054401329767256597203461858

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