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Properties

Label 2-460-115.103-c1-0-5
Degree $2$
Conductor $460$
Sign $0.766 + 0.642i$
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  + (−1.26 + 0.949i)3-s + (−2.13 − 0.673i)5-s + (0.0711 − 0.994i)7-s + (−0.138 + 0.470i)9-s + (0.617 − 0.961i)11-s + (3.39 − 0.243i)13-s + (3.34 − 1.17i)15-s + (1.77 − 4.76i)17-s + (1.26 − 2.76i)19-s + (0.853 + 1.32i)21-s + (−1.13 − 4.66i)23-s + (4.09 + 2.87i)25-s + (−1.93 − 5.18i)27-s + (6.85 − 3.13i)29-s + (1.12 + 7.82i)31-s + ⋯
L(s)  = 1  + (−0.732 + 0.548i)3-s + (−0.953 − 0.300i)5-s + (0.0268 − 0.375i)7-s + (−0.0460 + 0.156i)9-s + (0.186 − 0.289i)11-s + (0.942 − 0.0674i)13-s + (0.863 − 0.302i)15-s + (0.430 − 1.15i)17-s + (0.289 − 0.633i)19-s + (0.186 + 0.289i)21-s + (−0.236 − 0.971i)23-s + (0.818 + 0.574i)25-s + (−0.371 − 0.997i)27-s + (1.27 − 0.581i)29-s + (0.202 + 1.40i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804536 - 0.292824i\)
\(L(\frac12)\) \(\approx\) \(0.804536 - 0.292824i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.13 + 0.673i)T \)
23 \( 1 + (1.13 + 4.66i)T \)
good3 \( 1 + (1.26 - 0.949i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.0711 + 0.994i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (-0.617 + 0.961i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (-3.39 + 0.243i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-1.77 + 4.76i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-1.26 + 2.76i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-6.85 + 3.13i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (-1.12 - 7.82i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (0.503 + 0.921i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (-7.46 + 2.19i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (2.62 + 3.50i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (-2.09 + 2.09i)T - 47iT^{2} \)
53 \( 1 + (8.42 + 0.602i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (5.56 + 4.81i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (4.66 - 0.670i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (13.2 + 2.89i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (8.50 - 5.46i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-10.2 + 3.81i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (0.257 - 0.297i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-5.27 + 2.88i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-1.05 + 7.32i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-4.66 - 2.54i)T + (52.4 + 81.6i)T^{2} \)
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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

−10.92663860022271036919511814499, −10.39454434622386892350082328653, −9.117222894905771149620003946602, −8.268438093792182714314505921933, −7.30394564333878095144577478532, −6.20205666620847598572816905788, −5.00578503648025776503788270029, −4.33935547433881390713573096324, −3.09288036570935708718802164928, −0.68982149502343817661631230178, 1.29804342265130973537623212337, 3.26072797458930773933666901294, 4.28100536847949860101662093075, 5.81185089525337945177872593986, 6.35898135315535328748039189407, 7.50639340443585426325688979674, 8.236989551323811891198693922572, 9.324866026449575138630477423334, 10.55186278249246018514732113765, 11.30101559153770088831234603250

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