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Properties

Label 2-460-23.6-c1-0-0
Degree $2$
Conductor $460$
Sign $-0.207 - 0.978i$
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  + (−1.08 − 0.319i)3-s + (−0.841 + 0.540i)5-s + (0.0860 − 0.598i)7-s + (−1.44 − 0.927i)9-s + (0.638 + 1.39i)11-s + (0.955 + 6.64i)13-s + (1.08 − 0.319i)15-s + (−1.77 + 2.04i)17-s + (−0.883 − 1.01i)19-s + (−0.284 + 0.623i)21-s + (2.16 + 4.27i)23-s + (0.415 − 0.909i)25-s + (3.50 + 4.03i)27-s + (−4.67 + 5.40i)29-s + (−5.55 + 1.62i)31-s + ⋯
L(s)  = 1  + (−0.627 − 0.184i)3-s + (−0.376 + 0.241i)5-s + (0.0325 − 0.226i)7-s + (−0.480 − 0.309i)9-s + (0.192 + 0.421i)11-s + (0.265 + 1.84i)13-s + (0.280 − 0.0824i)15-s + (−0.430 + 0.496i)17-s + (−0.202 − 0.233i)19-s + (−0.0621 + 0.136i)21-s + (0.452 + 0.891i)23-s + (0.0830 − 0.181i)25-s + (0.673 + 0.777i)27-s + (−0.869 + 1.00i)29-s + (−0.996 + 0.292i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $-0.207 - 0.978i$
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: \((2,\ 460,\ (\ :1/2),\ -0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416689 + 0.514107i\)
\(L(\frac12)\) \(\approx\) \(0.416689 + 0.514107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.16 - 4.27i)T \)
good3 \( 1 + (1.08 + 0.319i)T + (2.52 + 1.62i)T^{2} \)
7 \( 1 + (-0.0860 + 0.598i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-0.638 - 1.39i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (-0.955 - 6.64i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (1.77 - 2.04i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (0.883 + 1.01i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (4.67 - 5.40i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (5.55 - 1.62i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (6.24 + 4.01i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (6.41 - 4.12i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (-12.2 - 3.59i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 - 1.94T + 47T^{2} \)
53 \( 1 + (1.59 - 11.1i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (1.88 + 13.0i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-5.45 + 1.60i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (0.362 - 0.794i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (-3.21 + 7.03i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (5.25 + 6.06i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.07 + 7.48i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (4.24 + 2.72i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (14.1 + 4.14i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (6.87 - 4.41i)T + (40.2 - 88.2i)T^{2} \)
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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

−11.19526073741554331744523017035, −10.84122623288527969639189393080, −9.315687516927523463137664030398, −8.840741308372315273208717164961, −7.33517028346248593815274119002, −6.77285594613620970828712757042, −5.77313517679348842130949690566, −4.53343989713657842210071551061, −3.51181887403008649588092454137, −1.72004039886468103183849081110, 0.44048216078010242182389498330, 2.65657531040828441500060452323, 3.96560645802709107943490321846, 5.33134690351391789727792010072, 5.75906111043009121593538619886, 7.11679611665467556083480295616, 8.226058633259834014131748130367, 8.812809410991417934449130964959, 10.16865716211173387557584435055, 10.87303736705195202358502258732

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