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Properties

Label 1-760-760.339-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.513 + 0.858i$
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  + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (−0.173 − 0.984i)21-s + (−0.939 + 0.342i)23-s + (0.5 + 0.866i)27-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + 37-s − 39-s + (0.766 − 0.642i)41-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (−0.5 + 0.866i)7-s + (0.173 − 0.984i)9-s + (−0.5 − 0.866i)11-s + (0.766 + 0.642i)13-s + (−0.173 − 0.984i)17-s + (−0.173 − 0.984i)21-s + (−0.939 + 0.342i)23-s + (0.5 + 0.866i)27-s + (−0.173 + 0.984i)29-s + (0.5 − 0.866i)31-s + (0.939 + 0.342i)33-s + 37-s − 39-s + (0.766 − 0.642i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.513 + 0.858i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.513 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4483597821 + 0.7903507931i\)
\(L(\frac12)\) \(\approx\) \(0.4483597821 + 0.7903507931i\)
\(L(1)\) \(\approx\) \(0.7210810264 + 0.2115959292i\)
\(L(1)\) \(\approx\) \(0.7210810264 + 0.2115959292i\)

Euler product

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

−22.24605098875268417926506122758, −21.10751091862960329059728473603, −20.225883186267290232741303648175, −19.498270558757494171163202721090, −18.64958385871306177570996808278, −17.652995498988567959039949012697, −17.3981630119187592424314603112, −16.23409151565913960123936844454, −15.70902247976167413899073710257, −14.475934255025410441886211185242, −13.44156347465827999057076964708, −12.892681949337525112527599449004, −12.22347993954060500909829045811, −11.01369266350031060148059313887, −10.48299937330604634200476959008, −9.65811648438673519660092032117, −8.10173574972657820662181781279, −7.63640214619462253302535159019, −6.48178203876427430121140308338, −6.00021442554967283646530852485, −4.74959575905133835234462784460, −3.88614540952396741168312324143, −2.500276816423421237904812811019, −1.34772778422372352289145194422, −0.3129038487283388362338395071, 0.8374469805871667978804411704, 2.4527715058657415568643062708, 3.48645856649650619405383526501, 4.45305744273411885526903080650, 5.638749601656674386222859418087, 5.99317652062991817837335627269, 7.082664358442009498737343668420, 8.44361155794849776896971418199, 9.22843273059753140861093557850, 9.947912124131493913719049447697, 11.0869937710253221888513855219, 11.542657178031760667985217726340, 12.4665012981779713242340667794, 13.411998472662316865953850018812, 14.36977149292476651519265413129, 15.48599387827755485784333510956, 16.08679563507125761809940035201, 16.46453502514672794967624243281, 17.71458986270573391435306076104, 18.424088146520876772482431301557, 19.010047127698489180138147361208, 20.267635842209318407497268680261, 21.069678293620060432393284261174, 21.78112905857109052219116102602, 22.3153460297333289466984117610

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