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Properties

Label 2-460-23.12-c1-0-3
Degree $2$
Conductor $460$
Sign $0.908 + 0.417i$
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  + (0.241 − 1.68i)3-s + (−0.959 + 0.281i)5-s + (2.58 + 2.97i)7-s + (0.105 + 0.0309i)9-s + (1.84 + 1.18i)11-s + (2.08 − 2.40i)13-s + (0.241 + 1.68i)15-s + (0.820 + 1.79i)17-s + (0.904 − 1.97i)19-s + (5.63 − 3.62i)21-s + (−3.74 − 3.00i)23-s + (0.841 − 0.540i)25-s + (2.19 − 4.80i)27-s + (0.693 + 1.51i)29-s + (0.122 + 0.852i)31-s + ⋯
L(s)  = 1  + (0.139 − 0.971i)3-s + (−0.429 + 0.125i)5-s + (0.975 + 1.12i)7-s + (0.0351 + 0.0103i)9-s + (0.555 + 0.357i)11-s + (0.577 − 0.666i)13-s + (0.0624 + 0.434i)15-s + (0.198 + 0.435i)17-s + (0.207 − 0.454i)19-s + (1.23 − 0.790i)21-s + (−0.779 − 0.625i)23-s + (0.168 − 0.108i)25-s + (0.422 − 0.925i)27-s + (0.128 + 0.281i)29-s + (0.0220 + 0.153i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.908 + 0.417i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.908 + 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58534 - 0.347140i\)
\(L(\frac12)\) \(\approx\) \(1.58534 - 0.347140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.959 - 0.281i)T \)
23 \( 1 + (3.74 + 3.00i)T \)
good3 \( 1 + (-0.241 + 1.68i)T + (-2.87 - 0.845i)T^{2} \)
7 \( 1 + (-2.58 - 2.97i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-1.84 - 1.18i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (-2.08 + 2.40i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-0.820 - 1.79i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (-0.904 + 1.97i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-0.693 - 1.51i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-0.122 - 0.852i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (-1.64 - 0.483i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (1.93 - 0.568i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (-1.39 + 9.72i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 6.47T + 47T^{2} \)
53 \( 1 + (-6.84 - 7.89i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (8.37 - 9.66i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (-1.26 - 8.81i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-11.5 + 7.44i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (-2.28 + 1.46i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (-0.192 + 0.420i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (6.61 - 7.63i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (7.69 + 2.25i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (0.0222 - 0.154i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (11.4 - 3.37i)T + (81.6 - 52.4i)T^{2} \)
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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

−11.19587199036063751634928925035, −10.20051375404337656163879030443, −8.831749879459785660148995847265, −8.245972928142294466178049109078, −7.43524609662377103357952464588, −6.42885824508704770850766081458, −5.40568004836901711124784893873, −4.16351538946117405878388632470, −2.57954713488230069143871390108, −1.42104772654170486659645160738, 1.37041394103026499374859778541, 3.60303011847313057117869357812, 4.16609132119277764226653604597, 5.06496109029840721791154693549, 6.52782648149568912687814334837, 7.62116474100822701090682389334, 8.396221081652963772310926875250, 9.498517428161243555351262837229, 10.15355313736086803972234431891, 11.27758597284201668995162986948

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