Properties

Label 2-460-23.2-c1-0-0
Degree $2$
Conductor $460$
Sign $0.115 - 0.993i$
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  + (−0.0800 − 0.557i)3-s + (−0.959 − 0.281i)5-s + (−2.98 + 3.44i)7-s + (2.57 − 0.755i)9-s + (−1.53 + 0.983i)11-s + (2.30 + 2.66i)13-s + (−0.0800 + 0.557i)15-s + (−2.29 + 5.02i)17-s + (0.276 + 0.604i)19-s + (2.15 + 1.38i)21-s + (0.619 + 4.75i)23-s + (0.841 + 0.540i)25-s + (−1.32 − 2.90i)27-s + (0.245 − 0.537i)29-s + (−0.421 + 2.93i)31-s + ⋯
L(s)  = 1  + (−0.0462 − 0.321i)3-s + (−0.429 − 0.125i)5-s + (−1.12 + 1.30i)7-s + (0.858 − 0.251i)9-s + (−0.461 + 0.296i)11-s + (0.640 + 0.739i)13-s + (−0.0206 + 0.143i)15-s + (−0.556 + 1.21i)17-s + (0.0633 + 0.138i)19-s + (0.470 + 0.302i)21-s + (0.129 + 0.991i)23-s + (0.168 + 0.108i)25-s + (−0.255 − 0.559i)27-s + (0.0455 − 0.0997i)29-s + (−0.0757 + 0.526i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.115 - 0.993i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.115 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707804 + 0.630353i\)
\(L(\frac12)\) \(\approx\) \(0.707804 + 0.630353i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.619 - 4.75i)T \)
good3 \( 1 + (0.0800 + 0.557i)T + (-2.87 + 0.845i)T^{2} \)
7 \( 1 + (2.98 - 3.44i)T + (-0.996 - 6.92i)T^{2} \)
11 \( 1 + (1.53 - 0.983i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (-2.30 - 2.66i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (2.29 - 5.02i)T + (-11.1 - 12.8i)T^{2} \)
19 \( 1 + (-0.276 - 0.604i)T + (-12.4 + 14.3i)T^{2} \)
29 \( 1 + (-0.245 + 0.537i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.421 - 2.93i)T + (-29.7 - 8.73i)T^{2} \)
37 \( 1 + (-9.01 + 2.64i)T + (31.1 - 20.0i)T^{2} \)
41 \( 1 + (8.05 + 2.36i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.932 - 6.48i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 - 0.201T + 47T^{2} \)
53 \( 1 + (1.38 - 1.60i)T + (-7.54 - 52.4i)T^{2} \)
59 \( 1 + (3.67 + 4.23i)T + (-8.39 + 58.3i)T^{2} \)
61 \( 1 + (1.79 - 12.5i)T + (-58.5 - 17.1i)T^{2} \)
67 \( 1 + (3.69 + 2.37i)T + (27.8 + 60.9i)T^{2} \)
71 \( 1 + (1.79 + 1.15i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (3.90 + 8.54i)T + (-47.8 + 55.1i)T^{2} \)
79 \( 1 + (-1.05 - 1.22i)T + (-11.2 + 78.1i)T^{2} \)
83 \( 1 + (1.80 - 0.529i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (1.01 + 7.03i)T + (-85.3 + 25.0i)T^{2} \)
97 \( 1 + (-10.2 - 3.02i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.44754227482559413033192976018, −10.27836623629897322325197832835, −9.374356329115555601182012335717, −8.683766760094105303725980978644, −7.56690129445640635669819781835, −6.51772285008025715006479818960, −5.86523820074306242068156733302, −4.40831292920008343096929978339, −3.28094223841978942424775145718, −1.81645931262125139306149258225, 0.59772386732264401498876535170, 2.95131733141122640522333791289, 3.93949849022967237030446369914, 4.87306107859697221101369802714, 6.37113358092152813785622624049, 7.15348412822545356533886242674, 7.959462697625308141200262876269, 9.232906124352002698333755908093, 10.17886478146614415274320638345, 10.61488762730947007351558274266

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