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Properties

Label 1-760-760.749-r1-0-0
Degree $1$
Conductor $760$
Sign $0.910 - 0.412i$
Analytic cond. $81.6733$
Root an. cond. $81.6733$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s − 27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s − 37-s + 39-s + (0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s − 7-s + (−0.5 + 0.866i)9-s − 11-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)21-s + (0.5 − 0.866i)23-s − 27-s + (−0.5 + 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s − 37-s + 39-s + (0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $0.910 - 0.412i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ 0.910 - 0.412i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.310807262 - 0.2833009999i\)
\(L(\frac12)\) \(\approx\) \(1.310807262 - 0.2833009999i\)
\(L(1)\) \(\approx\) \(0.9710385968 + 0.2098677764i\)
\(L(1)\) \(\approx\) \(0.9710385968 + 0.2098677764i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.46670438352946721077624984184, −21.20748996700456857697756714493, −20.67700313467758071187583276804, −19.682634119939081690654083691137, −18.9435340265088749280018676055, −18.518018119838567678920966250778, −17.568885123587428958071470869507, −16.50543954557694643575426152445, −15.789770076814916186803062430905, −14.880828850903232657505755632548, −13.716924996042223086868358936357, −13.42390522780995270419374332586, −12.49315543347652392852951197280, −11.72608106073364912740551581945, −10.66085681988380725523033493621, −9.45686300095972486694661043189, −9.00591269595578108258114824268, −7.732256109781810664623167921935, −7.18807248617088671585682566271, −6.21633249523879838648899628995, −5.36099342782981968835508606084, −3.83964671765430341589007170180, −3.00510332768134188906106732115, −2.09698567624128880570245607988, −0.828710617643057348544743751204, 0.35144437689266848053685456991, 2.12843583080840956660836852969, 3.23801486708686435831473501182, 3.68372985285614470272637021891, 5.08198849899188854060765557400, 5.72275810262918928400113260612, 6.95283071520902206562909942016, 8.085065019459574555023268894587, 8.731485528756691882734988189797, 9.76413441974804958795256670188, 10.44946229812268023072460945132, 11.00173051946962783635184961774, 12.60760371039988042070374103211, 13.01374121604651495061414646358, 14.03606379972397167684377291, 15.02122310231993523338690944962, 15.59012070056087331910700238305, 16.3658694250267978888030399419, 17.01518154306018469614331525432, 18.31241190558924850934728497853, 18.96962111948354890728315402040, 19.966199229575138139185402471330, 20.47642528308359800031512701080, 21.35405159010374286752127821436, 22.0744962931204875051978537119

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