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Properties

Label 2-460-23.16-c1-0-6
Degree $2$
Conductor $460$
Sign $0.342 + 0.939i$
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  + (2.44 − 1.57i)3-s + (−0.415 − 0.909i)5-s + (2.39 − 0.702i)7-s + (2.26 − 4.95i)9-s + (−1.36 − 1.57i)11-s + (−4.61 − 1.35i)13-s + (−2.44 − 1.57i)15-s + (−0.933 + 6.49i)17-s + (0.880 + 6.12i)19-s + (4.74 − 5.47i)21-s + (2.85 − 3.85i)23-s + (−0.654 + 0.755i)25-s + (−1.01 − 7.04i)27-s + (0.00773 − 0.0537i)29-s + (9.13 + 5.87i)31-s + ⋯
L(s)  = 1  + (1.41 − 0.907i)3-s + (−0.185 − 0.406i)5-s + (0.904 − 0.265i)7-s + (0.754 − 1.65i)9-s + (−0.412 − 0.475i)11-s + (−1.28 − 0.376i)13-s + (−0.631 − 0.405i)15-s + (−0.226 + 1.57i)17-s + (0.201 + 1.40i)19-s + (1.03 − 1.19i)21-s + (0.595 − 0.803i)23-s + (−0.130 + 0.151i)25-s + (−0.194 − 1.35i)27-s + (0.00143 − 0.00998i)29-s + (1.64 + 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79299 - 1.25469i\)
\(L(\frac12)\) \(\approx\) \(1.79299 - 1.25469i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.85 + 3.85i)T \)
good3 \( 1 + (-2.44 + 1.57i)T + (1.24 - 2.72i)T^{2} \)
7 \( 1 + (-2.39 + 0.702i)T + (5.88 - 3.78i)T^{2} \)
11 \( 1 + (1.36 + 1.57i)T + (-1.56 + 10.8i)T^{2} \)
13 \( 1 + (4.61 + 1.35i)T + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (0.933 - 6.49i)T + (-16.3 - 4.78i)T^{2} \)
19 \( 1 + (-0.880 - 6.12i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (-0.00773 + 0.0537i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-9.13 - 5.87i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-3.12 + 6.83i)T + (-24.2 - 27.9i)T^{2} \)
41 \( 1 + (1.35 + 2.97i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.58 - 2.94i)T + (17.8 - 39.1i)T^{2} \)
47 \( 1 + 3.24T + 47T^{2} \)
53 \( 1 + (-4.64 + 1.36i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (13.3 + 3.91i)T + (49.6 + 31.8i)T^{2} \)
61 \( 1 + (-8.36 - 5.37i)T + (25.3 + 55.4i)T^{2} \)
67 \( 1 + (-4.47 + 5.16i)T + (-9.53 - 66.3i)T^{2} \)
71 \( 1 + (-3.07 + 3.54i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-0.727 - 5.06i)T + (-70.0 + 20.5i)T^{2} \)
79 \( 1 + (-0.767 - 0.225i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (3.09 - 6.76i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (3.96 - 2.54i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-3.12 - 6.84i)T + (-63.5 + 73.3i)T^{2} \)
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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

−10.78504383479792226941854889461, −9.889986134431624556423779633436, −8.645419882224046046248690471135, −8.133097865292779973905067692018, −7.64217219210778317573550853040, −6.45419726747159054259722643163, −5.02247223963658862342108063312, −3.78590592464195129542474053562, −2.54277480338818249938427130645, −1.38448007456420114738403375694, 2.37287470455075272263396576484, 2.98592486803711836537543206202, 4.66523811769618704970123059099, 4.87008091753196613270341598502, 7.02104111548780464237563349882, 7.70484503786115747652634523770, 8.594269571021336380556206480213, 9.590622745564568544716453652483, 9.906748332819360505765969283666, 11.26718200350457909311242003461

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