Properties

Label 2-460-23.13-c1-0-2
Degree $2$
Conductor $460$
Sign $0.370 - 0.928i$
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  + (−1.58 − 1.02i)3-s + (−0.415 + 0.909i)5-s + (−0.0454 − 0.0133i)7-s + (0.235 + 0.515i)9-s + (−3.60 + 4.15i)11-s + (4.56 − 1.34i)13-s + (1.58 − 1.02i)15-s + (0.0975 + 0.678i)17-s + (−1.17 + 8.18i)19-s + (0.0586 + 0.0676i)21-s + (2.74 + 3.93i)23-s + (−0.654 − 0.755i)25-s + (−0.654 + 4.54i)27-s + (−0.572 − 3.98i)29-s + (5.28 − 3.39i)31-s + ⋯
L(s)  = 1  + (−0.917 − 0.589i)3-s + (−0.185 + 0.406i)5-s + (−0.0171 − 0.00504i)7-s + (0.0784 + 0.171i)9-s + (−1.08 + 1.25i)11-s + (1.26 − 0.371i)13-s + (0.410 − 0.263i)15-s + (0.0236 + 0.164i)17-s + (−0.270 + 1.87i)19-s + (0.0127 + 0.0147i)21-s + (0.571 + 0.820i)23-s + (−0.130 − 0.151i)25-s + (−0.125 + 0.875i)27-s + (−0.106 − 0.739i)29-s + (0.949 − 0.610i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.370 - 0.928i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (381, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614984 + 0.416942i\)
\(L(\frac12)\) \(\approx\) \(0.614984 + 0.416942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-2.74 - 3.93i)T \)
good3 \( 1 + (1.58 + 1.02i)T + (1.24 + 2.72i)T^{2} \)
7 \( 1 + (0.0454 + 0.0133i)T + (5.88 + 3.78i)T^{2} \)
11 \( 1 + (3.60 - 4.15i)T + (-1.56 - 10.8i)T^{2} \)
13 \( 1 + (-4.56 + 1.34i)T + (10.9 - 7.02i)T^{2} \)
17 \( 1 + (-0.0975 - 0.678i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.17 - 8.18i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.572 + 3.98i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-5.28 + 3.39i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-3.61 - 7.90i)T + (-24.2 + 27.9i)T^{2} \)
41 \( 1 + (4.21 - 9.22i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-9.16 - 5.88i)T + (17.8 + 39.1i)T^{2} \)
47 \( 1 + 7.74T + 47T^{2} \)
53 \( 1 + (-5.43 - 1.59i)T + (44.5 + 28.6i)T^{2} \)
59 \( 1 + (1.98 - 0.582i)T + (49.6 - 31.8i)T^{2} \)
61 \( 1 + (1.42 - 0.918i)T + (25.3 - 55.4i)T^{2} \)
67 \( 1 + (8.48 + 9.78i)T + (-9.53 + 66.3i)T^{2} \)
71 \( 1 + (0.937 + 1.08i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-1.14 + 7.97i)T + (-70.0 - 20.5i)T^{2} \)
79 \( 1 + (15.3 - 4.52i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (5.94 + 13.0i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.857 - 0.551i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.91 - 4.19i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.35154959105019790499896270495, −10.41472692122877817995487433686, −9.736247878549504782326162407522, −8.154692729683479034611368394367, −7.60279172448789484965432125789, −6.37419540293629171218665612490, −5.86348466635764890983593870867, −4.59319473529585271995195113530, −3.19905610558308195186825881722, −1.50400039440073461608106633897, 0.54381392553711000270001178451, 2.84288689851316222284047120195, 4.27050098646921616629090649972, 5.19746741576839783196013765266, 5.93632940789418313793464878972, 7.06050488402129717407071456203, 8.484101826707884803991183220133, 8.892288598962718475539049040889, 10.32686395504905975532773654411, 11.09431880917504544330610710025

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