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Properties

Label 2-460-23.4-c1-0-3
Degree $2$
Conductor $460$
Sign $0.901 - 0.432i$
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  + (−2.73 + 0.802i)3-s + (0.841 + 0.540i)5-s + (−0.359 − 2.49i)7-s + (4.30 − 2.76i)9-s + (−1.78 + 3.91i)11-s + (0.846 − 5.89i)13-s + (−2.73 − 0.802i)15-s + (3.87 + 4.46i)17-s + (−0.652 + 0.752i)19-s + (2.98 + 6.54i)21-s + (4.71 − 0.888i)23-s + (0.415 + 0.909i)25-s + (−3.96 + 4.57i)27-s + (6.29 + 7.27i)29-s + (10.1 + 2.98i)31-s + ⋯
L(s)  = 1  + (−1.57 + 0.463i)3-s + (0.376 + 0.241i)5-s + (−0.135 − 0.944i)7-s + (1.43 − 0.923i)9-s + (−0.539 + 1.18i)11-s + (0.234 − 1.63i)13-s + (−0.706 − 0.207i)15-s + (0.939 + 1.08i)17-s + (−0.149 + 0.172i)19-s + (0.652 + 1.42i)21-s + (0.982 − 0.185i)23-s + (0.0830 + 0.181i)25-s + (−0.762 + 0.880i)27-s + (1.16 + 1.35i)29-s + (1.82 + 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.901 - 0.432i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.901 - 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.847700 + 0.192652i\)
\(L(\frac12)\) \(\approx\) \(0.847700 + 0.192652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.71 + 0.888i)T \)
good3 \( 1 + (2.73 - 0.802i)T + (2.52 - 1.62i)T^{2} \)
7 \( 1 + (0.359 + 2.49i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (1.78 - 3.91i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (-0.846 + 5.89i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-3.87 - 4.46i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (0.652 - 0.752i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-6.29 - 7.27i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-10.1 - 2.98i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (2.30 - 1.48i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (2.00 + 1.29i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (5.14 - 1.50i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 - 6.36T + 47T^{2} \)
53 \( 1 + (1.21 + 8.48i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-0.0744 + 0.517i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (-4.06 - 1.19i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (-2.62 - 5.74i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (4.27 + 9.36i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.27 - 1.46i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (-2.19 + 15.2i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-2.04 + 1.31i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (7.32 - 2.15i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (-9.48 - 6.09i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.69252844791368416434681546757, −10.33686272036685861490823620094, −10.06243910258202229828295493594, −8.299536122140760367407321604605, −7.16537188692951500249584043379, −6.37244510706695310802792552277, −5.34300509009120410281639248943, −4.67810609877343367407013992918, −3.28405183756968195751076200735, −1.03031479686380236231662186609, 0.919798823520072136410791579738, 2.63751261659306538323102570192, 4.61160202261401238102419699891, 5.50570891077504505472075367010, 6.16463940373096089529660790293, 6.94264849750012568998147328591, 8.292809379026577410934638367166, 9.279030555591811397679869737051, 10.24369306570641508503754051959, 11.36466452070195332991191503817

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