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Properties

Label 2-460-115.22-c1-0-5
Degree $2$
Conductor $460$
Sign $0.443 + 0.896i$
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  + (−2.36 + 2.36i)3-s + (−2.19 − 0.429i)5-s + (−2.93 + 2.93i)7-s − 8.20i·9-s + 1.20i·11-s + (2.14 − 2.14i)13-s + (6.21 − 4.17i)15-s + (−3.74 + 3.74i)17-s + 5.51·19-s − 13.8i·21-s + (−0.475 − 4.77i)23-s + (4.63 + 1.88i)25-s + (12.3 + 12.3i)27-s − 1.52i·29-s − 5.73·31-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)3-s + (−0.981 − 0.191i)5-s + (−1.10 + 1.10i)7-s − 2.73i·9-s + 0.363i·11-s + (0.595 − 0.595i)13-s + (1.60 − 1.07i)15-s + (−0.909 + 0.909i)17-s + 1.26·19-s − 3.02i·21-s + (−0.0990 − 0.995i)23-s + (0.926 + 0.376i)25-s + (2.37 + 2.37i)27-s − 0.283i·29-s − 1.02·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(3.67311\)
Root analytic conductor: \(1.91653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{460} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 460,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.129133 - 0.0801646i\)
\(L(\frac12)\) \(\approx\) \(0.129133 - 0.0801646i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 0.429i)T \)
23 \( 1 + (0.475 + 4.77i)T \)
good3 \( 1 + (2.36 - 2.36i)T - 3iT^{2} \)
7 \( 1 + (2.93 - 2.93i)T - 7iT^{2} \)
11 \( 1 - 1.20iT - 11T^{2} \)
13 \( 1 + (-2.14 + 2.14i)T - 13iT^{2} \)
17 \( 1 + (3.74 - 3.74i)T - 17iT^{2} \)
19 \( 1 - 5.51T + 19T^{2} \)
29 \( 1 + 1.52iT - 29T^{2} \)
31 \( 1 + 5.73T + 31T^{2} \)
37 \( 1 + (-3.78 + 3.78i)T - 37iT^{2} \)
41 \( 1 + 0.563T + 41T^{2} \)
43 \( 1 + (5.82 + 5.82i)T + 43iT^{2} \)
47 \( 1 + (4.05 + 4.05i)T + 47iT^{2} \)
53 \( 1 + (1.04 + 1.04i)T + 53iT^{2} \)
59 \( 1 + 0.826iT - 59T^{2} \)
61 \( 1 - 0.187iT - 61T^{2} \)
67 \( 1 + (-6.63 + 6.63i)T - 67iT^{2} \)
71 \( 1 + 4.26T + 71T^{2} \)
73 \( 1 + (3.06 - 3.06i)T - 73iT^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + (1.42 + 1.42i)T + 83iT^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (-3.19 + 3.19i)T - 97iT^{2} \)
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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

−10.95083749027954954004128366812, −10.11169864978994179567831605122, −9.271428739111797764846561211967, −8.514483863999755714804068289229, −6.85642908598079634888289586869, −5.97440893328837728209348267780, −5.20539390434922614296712962371, −4.09769241512777544304360847506, −3.26667447257578807888117157749, −0.13262742271140096849271290417, 1.13661227501798933223750659549, 3.21867377384659519691407495800, 4.56823784821098565688063036380, 5.83414763205283648396365969318, 6.91475875907023063002901588620, 7.09140978743544850208078486848, 8.067999933508677777688548652547, 9.565272267243763544218068843803, 10.77470636563801870296814915215, 11.40689958333566959901850691231

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