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Properties

Label 1-760-760.93-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.128 + 0.991i$
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  + (0.984 − 0.173i)3-s + (0.866 + 0.5i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (0.939 + 0.342i)21-s + (−0.642 + 0.766i)23-s + (0.866 − 0.5i)27-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (0.642 + 0.766i)33-s − i·37-s − 39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (0.984 − 0.173i)3-s + (0.866 + 0.5i)7-s + (0.939 − 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 − 0.173i)13-s + (−0.342 + 0.939i)17-s + (0.939 + 0.342i)21-s + (−0.642 + 0.766i)23-s + (0.866 − 0.5i)27-s + (−0.939 + 0.342i)29-s + (−0.5 + 0.866i)31-s + (0.642 + 0.766i)33-s − i·37-s − 39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.128 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.781259005 + 2.026741517i\)
\(L(\frac12)\) \(\approx\) \(1.781259005 + 2.026741517i\)
\(L(1)\) \(\approx\) \(1.483536822 + 0.3234367420i\)
\(L(1)\) \(\approx\) \(1.483536822 + 0.3234367420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (-0.342 + 0.939i)T \)
23 \( 1 + (-0.642 + 0.766i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 + (-0.642 - 0.766i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (-0.342 - 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
73 \( 1 + (-0.984 + 0.173i)T \)
79 \( 1 + (-0.173 - 0.984i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.342 + 0.939i)T \)
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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

−21.83920011611133695475249463851, −21.12255735800008018430364211795, −20.27418171137935138311695992178, −19.80403538932420933678379689357, −18.81364123253481077506389790778, −18.132967744244973447255663489341, −16.97492640538566668526352844363, −16.3714701296491687147381595763, −15.26907037026351412577042521016, −14.49200742017279617732106918942, −13.98960275910318914949277550542, −13.218175838094515179101400083245, −12.04306190682531580992719747684, −11.18108632504908017686445103180, −10.26527993204427299102638293040, −9.31304109439674505912837925072, −8.62508028614084094670444711668, −7.634294029960210149983302192431, −7.09203806852108384099335236361, −5.66678190972186508010457548057, −4.521191836562251976567355474623, −3.89356864554098551017601655131, −2.66783003359886126902030908101, −1.82033709824723855531403112363, −0.47627960924019079717962521333, 1.56226245122343193722145523952, 2.03294307177652210597763688951, 3.25528603680213563470792006992, 4.3092860566459810533199468862, 5.137641348364465306546884626812, 6.46384016465553926807171751098, 7.47367722722609043349042603401, 8.07463223285663939855154444061, 9.05211473895817424209002383148, 9.71461946376203592285965903178, 10.73039799166551004107915794245, 11.95305164426342824830350812352, 12.50384507683018997916711898053, 13.48773120850002216056001014902, 14.49365439546105607922174865125, 14.925037284362370221777671227734, 15.55218582980464195167148950949, 16.904798566118433672881420797541, 17.71607357523205241854544719776, 18.375110504115473080650016233758, 19.40415998124243060133913043038, 19.98841028209133509466696600945, 20.66518209357750326190405817737, 21.70860617192290510767942553652, 22.064978344350585166769971305269

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