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Properties

Label 2-460-23.6-c1-0-4
Degree $2$
Conductor $460$
Sign $0.928 - 0.372i$
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  + (2.68 + 0.789i)3-s + (0.841 − 0.540i)5-s + (−0.0887 + 0.617i)7-s + (4.08 + 2.62i)9-s + (−0.0756 − 0.165i)11-s + (−0.108 − 0.756i)13-s + (2.68 − 0.789i)15-s + (−2.33 + 2.69i)17-s + (−0.396 − 0.457i)19-s + (−0.725 + 1.58i)21-s + (3.84 − 2.87i)23-s + (0.415 − 0.909i)25-s + (3.41 + 3.93i)27-s + (1.73 − 2.00i)29-s + (−4.00 + 1.17i)31-s + ⋯
L(s)  = 1  + (1.55 + 0.456i)3-s + (0.376 − 0.241i)5-s + (−0.0335 + 0.233i)7-s + (1.36 + 0.875i)9-s + (−0.0227 − 0.0499i)11-s + (−0.0301 − 0.209i)13-s + (0.694 − 0.203i)15-s + (−0.565 + 0.652i)17-s + (−0.0908 − 0.104i)19-s + (−0.158 + 0.346i)21-s + (0.800 − 0.598i)23-s + (0.0830 − 0.181i)25-s + (0.656 + 0.758i)27-s + (0.322 − 0.372i)29-s + (−0.719 + 0.211i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.928 - 0.372i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.928 - 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33544 + 0.451133i\)
\(L(\frac12)\) \(\approx\) \(2.33544 + 0.451133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-3.84 + 2.87i)T \)
good3 \( 1 + (-2.68 - 0.789i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (0.0887 - 0.617i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (0.0756 + 0.165i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.108 + 0.756i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (2.33 - 2.69i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.396 + 0.457i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (-1.73 + 2.00i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (4.00 - 1.17i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (7.97 + 5.12i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (9.42 - 6.05i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (1.64 + 0.484i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 4.54T + 47T^{2} \)
53 \( 1 + (-0.279 + 1.94i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.381 - 2.65i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (4.25 - 1.25i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (-0.574 + 1.25i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-2.59 + 5.68i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (-3.68 - 4.25i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.73 + 12.0i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (7.36 + 4.73i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-11.8 - 3.48i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (12.2 - 7.85i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.77192819894021349095707955790, −10.08890139919710636442676616232, −9.007493138561864999622345830716, −8.729302137351862109805242412193, −7.73025194253623329936415173445, −6.59076581089388556080757824339, −5.21142807722519968169585400069, −4.08959858261651571302073781604, −3.01983781650971787985126879721, −1.94211655910358561109462677456, 1.70658867695013178973809237037, 2.81774077698484909672463637944, 3.78832449104842698033876122524, 5.22123923942049734588852993741, 6.81027429184610824678192227860, 7.27393245022024227530856960096, 8.428866918236112494422676350162, 9.057747425514551291188252145556, 9.871302965968952949725785721625, 10.86613371986633154224618033666

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