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Properties

Label 2-460-115.22-c1-0-10
Degree $2$
Conductor $460$
Sign $0.512 + 0.858i$
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  + (1.09 − 1.09i)3-s + (1.88 − 1.20i)5-s + (2.67 − 2.67i)7-s + 0.606i·9-s − 2.02i·11-s + (−3.32 + 3.32i)13-s + (0.749 − 3.37i)15-s + (−2.17 + 2.17i)17-s − 0.910·19-s − 5.84i·21-s + (−2.63 + 4.00i)23-s + (2.11 − 4.53i)25-s + (3.94 + 3.94i)27-s − 3.41i·29-s + 1.18·31-s + ⋯
L(s)  = 1  + (0.631 − 0.631i)3-s + (0.843 − 0.537i)5-s + (1.00 − 1.00i)7-s + 0.202i·9-s − 0.611i·11-s + (−0.921 + 0.921i)13-s + (0.193 − 0.871i)15-s + (−0.527 + 0.527i)17-s − 0.208·19-s − 1.27i·21-s + (−0.549 + 0.835i)23-s + (0.423 − 0.906i)25-s + (0.759 + 0.759i)27-s − 0.634i·29-s + 0.213·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(460\)    =    \(2^{2} \cdot 5 \cdot 23\)
Sign: $0.512 + 0.858i$
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: \((2,\ 460,\ (\ :1/2),\ 0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77386 - 1.00645i\)
\(L(\frac12)\) \(\approx\) \(1.77386 - 1.00645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 1.20i)T \)
23 \( 1 + (2.63 - 4.00i)T \)
good3 \( 1 + (-1.09 + 1.09i)T - 3iT^{2} \)
7 \( 1 + (-2.67 + 2.67i)T - 7iT^{2} \)
11 \( 1 + 2.02iT - 11T^{2} \)
13 \( 1 + (3.32 - 3.32i)T - 13iT^{2} \)
17 \( 1 + (2.17 - 2.17i)T - 17iT^{2} \)
19 \( 1 + 0.910T + 19T^{2} \)
29 \( 1 + 3.41iT - 29T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + (0.268 - 0.268i)T - 37iT^{2} \)
41 \( 1 - 3.45T + 41T^{2} \)
43 \( 1 + (-2.89 - 2.89i)T + 43iT^{2} \)
47 \( 1 + (0.714 + 0.714i)T + 47iT^{2} \)
53 \( 1 + (7.75 + 7.75i)T + 53iT^{2} \)
59 \( 1 - 8.22iT - 59T^{2} \)
61 \( 1 - 5.34iT - 61T^{2} \)
67 \( 1 + (-7.39 + 7.39i)T - 67iT^{2} \)
71 \( 1 - 0.769T + 71T^{2} \)
73 \( 1 + (10.5 - 10.5i)T - 73iT^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + (6.32 + 6.32i)T + 83iT^{2} \)
89 \( 1 + 3.14T + 89T^{2} \)
97 \( 1 + (2.21 - 2.21i)T - 97iT^{2} \)
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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

−10.89444746117149040832915873889, −9.969898287673666605657254980299, −8.977649413930510177435100862942, −8.121361286999959771351525945040, −7.43382070103474623869109016698, −6.35035037550876861239730636131, −5.07252887007941644300456125228, −4.16511057348742892130984954047, −2.34160615078773174324546531700, −1.43172929531705802139805751987, 2.14416912218276570840109393759, 2.92394141879690735260170605788, 4.53641210664034945472168247404, 5.37788774360322158933204420997, 6.50285094734949145831022633528, 7.69873693902756537004525703381, 8.698294444974440547879879036283, 9.429106522046607890182732749020, 10.15949958195764706180289391821, 11.01402222299403520710036106957

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