Properties

Label 2-460-23.4-c1-0-2
Degree $2$
Conductor $460$
Sign $0.218 - 0.975i$
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  + (0.0447 − 0.0131i)3-s + (0.841 + 0.540i)5-s + (0.423 + 2.94i)7-s + (−2.52 + 1.62i)9-s + (0.116 − 0.254i)11-s + (−0.287 + 1.99i)13-s + (0.0447 + 0.0131i)15-s + (1.46 + 1.69i)17-s + (−2.64 + 3.04i)19-s + (0.0577 + 0.126i)21-s + (2.21 − 4.25i)23-s + (0.415 + 0.909i)25-s + (−0.183 + 0.211i)27-s + (4.42 + 5.10i)29-s + (−3.52 − 1.03i)31-s + ⋯
L(s)  = 1  + (0.0258 − 0.00758i)3-s + (0.376 + 0.241i)5-s + (0.160 + 1.11i)7-s + (−0.840 + 0.540i)9-s + (0.0349 − 0.0766i)11-s + (−0.0796 + 0.553i)13-s + (0.0115 + 0.00339i)15-s + (0.356 + 0.411i)17-s + (−0.606 + 0.699i)19-s + (0.0125 + 0.0275i)21-s + (0.462 − 0.886i)23-s + (0.0830 + 0.181i)25-s + (−0.0352 + 0.0407i)27-s + (0.821 + 0.947i)29-s + (−0.632 − 0.185i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.218 - 0.975i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.218 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04055 + 0.833717i\)
\(L(\frac12)\) \(\approx\) \(1.04055 + 0.833717i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 4.25i)T \)
good3 \( 1 + (-0.0447 + 0.0131i)T + (2.52 - 1.62i)T^{2} \)
7 \( 1 + (-0.423 - 2.94i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-0.116 + 0.254i)T + (-7.20 - 8.31i)T^{2} \)
13 \( 1 + (0.287 - 1.99i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-1.46 - 1.69i)T + (-2.41 + 16.8i)T^{2} \)
19 \( 1 + (2.64 - 3.04i)T + (-2.70 - 18.8i)T^{2} \)
29 \( 1 + (-4.42 - 5.10i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (3.52 + 1.03i)T + (26.0 + 16.7i)T^{2} \)
37 \( 1 + (-5.26 + 3.38i)T + (15.3 - 33.6i)T^{2} \)
41 \( 1 + (-7.90 - 5.08i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-1.04 + 0.306i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + 4.44T + 47T^{2} \)
53 \( 1 + (0.0276 + 0.192i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (1.27 - 8.87i)T + (-56.6 - 16.6i)T^{2} \)
61 \( 1 + (2.28 + 0.670i)T + (51.3 + 32.9i)T^{2} \)
67 \( 1 + (4.17 + 9.14i)T + (-43.8 + 50.6i)T^{2} \)
71 \( 1 + (5.39 + 11.8i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-8.19 + 9.45i)T + (-10.3 - 72.2i)T^{2} \)
79 \( 1 + (0.917 - 6.37i)T + (-75.7 - 22.2i)T^{2} \)
83 \( 1 + (-7.09 + 4.55i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-0.984 + 0.289i)T + (74.8 - 48.1i)T^{2} \)
97 \( 1 + (10.8 + 6.94i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.16315105526023383932572494748, −10.50262221984742936829159159495, −9.280494065924265469202882671280, −8.648818550462587075142055906001, −7.74915069311706280034305672804, −6.35755843276970517112884144129, −5.72050680805973398360903989124, −4.62813799732868835268475051345, −3.01243130770561850171266073086, −2.00413836661084290101875823207, 0.844313819047024658433056171785, 2.72922462932695997408677178541, 3.97921567671417376860503855551, 5.13504673571800032785223623566, 6.16845089227265526068085795009, 7.20816807348607528326148663036, 8.126927699989528097549670221934, 9.149223177112084414228294812502, 9.940626427944202013189323365190, 10.89569166169854068405609392474

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