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Properties

Label 2-480-40.29-c1-0-6
Degree $2$
Conductor $480$
Sign $0.999 + 0.0190i$
Analytic cond. $3.83281$
Root an. cond. $1.95775$
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  + 3-s + (1.86 − 1.23i)5-s + 0.746i·7-s + 9-s + 5.36i·11-s + 2.92·13-s + (1.86 − 1.23i)15-s − 2.13i·17-s − 1.73i·19-s + 0.746i·21-s − 7.49i·23-s + (1.92 − 4.61i)25-s + 27-s + 6.74i·29-s − 2.64·31-s + ⋯
L(s)  = 1  + 0.577·3-s + (0.832 − 0.554i)5-s + 0.282i·7-s + 0.333·9-s + 1.61i·11-s + 0.811·13-s + (0.480 − 0.320i)15-s − 0.517i·17-s − 0.397i·19-s + 0.162i·21-s − 1.56i·23-s + (0.385 − 0.922i)25-s + 0.192·27-s + 1.25i·29-s − 0.475·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0190i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(480\)    =    \(2^{5} \cdot 3 \cdot 5\)
Sign: $0.999 + 0.0190i$
Analytic conductor: \(3.83281\)
Root analytic conductor: \(1.95775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{480} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 480,\ (\ :1/2),\ 0.999 + 0.0190i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99223 - 0.0190053i\)
\(L(\frac12)\) \(\approx\) \(1.99223 - 0.0190053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + (-1.86 + 1.23i)T \)
good7 \( 1 - 0.746iT - 7T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 - 2.92T + 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 7.49iT - 23T^{2} \)
29 \( 1 - 6.74iT - 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 1.07T + 37T^{2} \)
41 \( 1 + 11.2T + 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 - 1.73iT - 47T^{2} \)
53 \( 1 + 7.72T + 53T^{2} \)
59 \( 1 + 6.85iT - 59T^{2} \)
61 \( 1 - 6.45iT - 61T^{2} \)
67 \( 1 + 7.44T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 0.690iT - 73T^{2} \)
79 \( 1 - 2.64T + 79T^{2} \)
83 \( 1 + 5.85T + 83T^{2} \)
89 \( 1 + 7.59T + 89T^{2} \)
97 \( 1 - 14.1iT - 97T^{2} \)
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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

−10.75949310934502174141324926729, −9.971175087163534377442901810758, −9.119185390375187389277182408590, −8.572450041352674332761575449421, −7.31084924276855828693847288893, −6.43413076246727388936185137956, −5.16615964760967576757167229688, −4.33486721493403965482021699712, −2.72094406423078136074997532728, −1.61786609385062618659483279694, 1.54317322712221154025089766567, 3.05054885240179095965044996540, 3.85337945841723048975086837887, 5.63694126729536772182284383308, 6.19639261857522043618485356550, 7.41910059242633624251031526406, 8.378390871922745750954535949168, 9.178460801188512055515215830558, 10.12621495309831040779080376540, 10.89215815843214223710256276837

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