↑

Properties

Label 1-1480-1480.1013-r0-0-0
Degree $1$
Conductor $1480$
Sign $0.956 + 0.292i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)21-s + 23-s − i·27-s − i·29-s + i·31-s + (−0.866 − 0.5i)33-s + (−0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 + 0.866i)9-s + 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 − 0.866i)21-s + 23-s − i·27-s − i·29-s + i·31-s + (−0.866 − 0.5i)33-s + (−0.866 + 0.5i)39-s + (0.5 − 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.528411941 + 0.2281458476i\)
\(L(\frac12)\) \(\approx\) \(1.528411941 + 0.2281458476i\)
\(L(1)\) \(\approx\) \(1.061154310 + 0.004121954487i\)
\(L(1)\) \(\approx\) \(1.061154310 + 0.004121954487i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + T \)
29 \( 1 - iT \)
31 \( 1 + iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + (-0.866 + 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 - T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.76870667507475104217079723898, −20.09312008697146165149639257401, −18.947753057282284951304530616958, −18.30271981807943277519592598739, −17.40127219620024916778443422137, −16.96525798528497604362993963232, −16.25106055920366269439964355607, −15.44833513267741082699922615459, −14.543371359415273723482387773046, −13.95408129604859592447246897744, −13.01118355459352511644150172749, −11.796822655449860554582960178011, −11.4671275239232260857323704424, −10.95044058165944197818898074373, −9.63221043915224870907562661659, −9.417188469622858045384694169799, −8.17929931120104632545924921189, −7.16301310200607041759631850532, −6.5368935728580771981346252037, −5.55400476834559331554123080005, −4.67905385499683018336090646550, −4.152596160885551690732397682718, −3.12338429767164742589358062471, −1.57897468940679726936825905161, −0.82496708031553629113723122917, 1.20066772572344557210358534737, 1.54425404759790024451043144662, 2.99679320230629991743602460684, 4.0378264632685570831335549706, 5.21744362050639614943374593491, 5.571001097549420331381698493905, 6.55907274309520826522840627308, 7.36743948593240244314937266862, 8.23923482717228680500237298225, 8.935308269876505526358084704206, 10.17320150809128235387150829682, 10.854767057913641039209648942633, 11.581945380386320521432234044402, 12.27059034362016909125204779069, 12.85185366856780409609020943217, 13.90981672863264295974544465644, 14.615251651537326848079056197756, 15.44045714407845492015659227804, 16.342667631358172034383919803624, 17.09641887623578455517896791754, 17.70164836630732305609757551458, 18.338761609643452863471291008822, 19.01747005453940935354753583482, 19.88898926914980309058977228161, 20.74544553624171176497741329013

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