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Properties

Label 2-460-115.103-c1-0-10
Degree $2$
Conductor $460$
Sign $-0.108 + 0.994i$
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  + (1.82 − 1.36i)3-s + (−2.23 + 0.0541i)5-s + (0.160 − 2.24i)7-s + (0.621 − 2.11i)9-s + (2.67 − 4.16i)11-s + (−4.18 + 0.299i)13-s + (−4.00 + 3.15i)15-s + (0.285 − 0.765i)17-s + (1.10 − 2.41i)19-s + (−2.77 − 4.32i)21-s + (4.21 − 2.29i)23-s + (4.99 − 0.241i)25-s + (0.632 + 1.69i)27-s + (−4.75 + 2.17i)29-s + (−1.00 − 6.97i)31-s + ⋯
L(s)  = 1  + (1.05 − 0.789i)3-s + (−0.999 + 0.0242i)5-s + (0.0607 − 0.848i)7-s + (0.207 − 0.705i)9-s + (0.807 − 1.25i)11-s + (−1.16 + 0.0830i)13-s + (−1.03 + 0.814i)15-s + (0.0692 − 0.185i)17-s + (0.252 − 0.553i)19-s + (−0.606 − 0.943i)21-s + (0.878 − 0.477i)23-s + (0.998 − 0.0483i)25-s + (0.121 + 0.326i)27-s + (−0.882 + 0.403i)29-s + (−0.180 − 1.25i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.108 + 0.994i$
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: \((2,\ 460,\ (\ :1/2),\ -0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06200 - 1.18406i\)
\(L(\frac12)\) \(\approx\) \(1.06200 - 1.18406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0541i)T \)
23 \( 1 + (-4.21 + 2.29i)T \)
good3 \( 1 + (-1.82 + 1.36i)T + (0.845 - 2.87i)T^{2} \)
7 \( 1 + (-0.160 + 2.24i)T + (-6.92 - 0.996i)T^{2} \)
11 \( 1 + (-2.67 + 4.16i)T + (-4.56 - 10.0i)T^{2} \)
13 \( 1 + (4.18 - 0.299i)T + (12.8 - 1.85i)T^{2} \)
17 \( 1 + (-0.285 + 0.765i)T + (-12.8 - 11.1i)T^{2} \)
19 \( 1 + (-1.10 + 2.41i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (4.75 - 2.17i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (1.00 + 6.97i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-5.71 - 10.4i)T + (-20.0 + 31.1i)T^{2} \)
41 \( 1 + (4.11 - 1.20i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-4.06 - 5.43i)T + (-12.1 + 41.2i)T^{2} \)
47 \( 1 + (1.70 - 1.70i)T - 47iT^{2} \)
53 \( 1 + (7.08 + 0.506i)T + (52.4 + 7.54i)T^{2} \)
59 \( 1 + (3.37 + 2.92i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-12.5 + 1.80i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (-14.3 - 3.12i)T + (60.9 + 27.8i)T^{2} \)
71 \( 1 + (-0.135 + 0.0869i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (1.10 - 0.412i)T + (55.1 - 47.8i)T^{2} \)
79 \( 1 + (1.01 - 1.16i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (10.7 - 5.87i)T + (44.8 - 69.8i)T^{2} \)
89 \( 1 + (-1.44 + 10.0i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.32 - 1.26i)T + (52.4 + 81.6i)T^{2} \)
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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

−11.13633792462205989123163343969, −9.713161485545672958847115830172, −8.790877342721155563375019276929, −7.979081784608838918232444295524, −7.34125522041773360410232918072, −6.57625028474338378032964955229, −4.82016728395873813998399015868, −3.65439888176846225133644676183, −2.75041887881642472795104382396, −0.925255946140276062308544777178, 2.22916897150044089284715936486, 3.45386543915644824733815050966, 4.30405340884155792251539526414, 5.30974462078017168293102004230, 7.01530026036886906550994678474, 7.73807773526030881770382541904, 8.810116135463122033829137474174, 9.348591013593423402300864216794, 10.13128713106950515775187648270, 11.35955594016653603237677947494

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