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Properties

Label 1-760-760.619-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.671 + 0.740i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.671 + 0.740i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.671 + 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083371243 + 2.444512287i\)
\(L(\frac12)\) \(\approx\) \(1.083371243 + 2.444512287i\)
\(L(1)\) \(\approx\) \(1.253236195 + 0.6747334320i\)
\(L(1)\) \(\approx\) \(1.253236195 + 0.6747334320i\)

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

−21.969886153316328847771651776973, −20.90268378103941565144010051128, −20.12333328363680580804670787742, −19.71496096228397111333021377888, −18.46249222678983685999054047657, −18.09272011221747421560910513192, −17.22507728642477760341453266239, −16.37151814179798940002867464693, −14.96601211117103976901173283269, −14.54693362671281941254703179369, −13.86350596314504120933929873452, −12.81328881159864292260668527079, −12.04584124971633623809052464340, −11.40393468382208089579925444823, −10.22521667475184785481826169142, −9.15403782831994206057219801899, −8.36787800750783058932628497851, −7.58827392749894451331932946441, −6.83791225623663239641091496979, −5.74769670158128196486636193865, −4.744640962931198877978616176574, −3.53212384074078471998599292320, −2.508053339869431000473862259516, −1.50481184035874784298402007112, −0.557328899338605542899021140713, 1.41404842552821855961222028014, 2.304770316932569582344705346264, 3.69961100460156123329728795809, 4.28550243704398538718712252991, 5.220011159402220147045151139602, 6.275013267436570020273011040032, 7.5945606525354970757955845496, 8.26666481282820084519050222431, 9.27223876740147955516745050872, 9.81635652354755921817831736520, 11.008865158326579095013993762643, 11.53082352065025917713423315498, 12.55804992578038881046125147960, 13.98467409149788162346235298563, 14.299147526579214309805431916551, 15.051776926281850178443118496919, 15.92024401865386320263700233552, 17.00147112122637071753460517356, 17.28490653226167861904067457121, 18.60454628589231516985573218519, 19.51108344278350719609091760255, 20.06301660365770633744876253150, 21.07497734919921137183832760181, 21.60818515760888629672269191622, 22.1772774062475949042586588443

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