↑

Properties

Label 1-760-760.293-r0-0-0
Degree $1$
Conductor $760$
Sign $-0.830 - 0.557i$
Analytic cond. $3.52942$
Root an. cond. $3.52942$
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 − i·7-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (−0.866 + 0.5i)33-s − i·37-s − 39-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s − i·7-s + (0.5 − 0.866i)9-s − 11-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)21-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 31-s + (−0.866 + 0.5i)33-s − i·37-s − 39-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.830 - 0.557i$
Analytic conductor: \(3.52942\)
Root analytic conductor: \(3.52942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (0:\ ),\ -0.830 - 0.557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3447775743 - 1.131553059i\)
\(L(\frac12)\) \(\approx\) \(0.3447775743 - 1.131553059i\)
\(L(1)\) \(\approx\) \(0.9949010368 - 0.4942926104i\)
\(L(1)\) \(\approx\) \(0.9949010368 - 0.4942926104i\)

Euler product

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

−22.33026406813193350872679287343, −21.88649963534907843751053324852, −21.20111218062436044961807818264, −20.28089658690331054152747110589, −19.664355224529053762315423485071, −18.67075512521047513189908557610, −18.18134655919387196170869888147, −16.96388798671680956404489160471, −15.82271475813851313468728452465, −15.59108092426760238580922894870, −14.60678693315063616080668286628, −13.87349626588912172347864954361, −12.95504916027566889200478802204, −12.105460809180730700541661582495, −11.06768868378479262480274794290, −10.09331190486967818938712577580, −9.32672405091380521601705058745, −8.6120130129740262426250508822, −7.753074336960502213704807429319, −6.8056109619202765545392292371, −5.38797667475817071553136082679, −4.81544482209350207574594191497, −3.61001932513694091777220803746, −2.53550090337567821506350258435, −2.01864929887238291002118664353, 0.4426096937966065867470371425, 1.92210209858526315697390143000, 2.74115415256602762934962404877, 3.84761255571361043606673405025, 4.70700516187075998071537101278, 6.06239057542802206819299610252, 7.08097944808003213779330645155, 7.77778036308314507684515008876, 8.41358905253800485343073147567, 9.65005082657938156062585920193, 10.28300640717824428395501217275, 11.250257006428188018083453458100, 12.60909156262785178388060126564, 12.994026270676652348680747946661, 13.89971639010086249679288467535, 14.60124093547784427349108995770, 15.46070033574133736287362914803, 16.31811405160076797243116189877, 17.49444551831663694259469893067, 17.97836700605118400041632794966, 18.99707119007710548553932069257, 19.85540133246853261839494598808, 20.22788256779962517986287911710, 21.109006203687137831355527171586, 22.003762379462338686189156182844

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