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Properties

Label 1-760-760.739-r1-0-0
Degree $1$
Conductor $760$
Sign $-0.776 - 0.630i$
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  + (−0.173 − 0.984i)3-s + (−0.5 + 0.866i)7-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.5 + 0.866i)7-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (0.173 − 0.984i)13-s + (0.939 + 0.342i)17-s + (0.939 + 0.342i)21-s + (0.766 + 0.642i)23-s + (0.5 + 0.866i)27-s + (0.939 − 0.342i)29-s + (0.5 − 0.866i)31-s + (−0.766 + 0.642i)33-s + 37-s − 39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.776 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(760\)    =    \(2^{3} \cdot 5 \cdot 19\)
Sign: $-0.776 - 0.630i$
Analytic conductor: \(81.6733\)
Root analytic conductor: \(81.6733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{760} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 760,\ (1:\ ),\ -0.776 - 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3944656954 - 1.111027052i\)
\(L(\frac12)\) \(\approx\) \(0.3944656954 - 1.111027052i\)
\(L(1)\) \(\approx\) \(0.8378230927 - 0.3300919541i\)
\(L(1)\) \(\approx\) \(0.8378230927 - 0.3300919541i\)

Euler product

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

−22.72700207062358478083629187113, −21.53882621711035677440983213984, −21.00588593476273832738472089642, −20.237714686193469152816549538383, −19.501106972900327310060235197160, −18.45368936950662067065116363664, −17.48196663401456050996730641122, −16.63035335158724832929320295807, −16.2184593897995096351911277160, −15.26048186308238019795022042161, −14.39979015172438732293432951750, −13.6955816709394752869038274748, −12.5588860982131195725078677280, −11.733673505264065624432709998111, −10.66748956808420789045490099964, −10.116776385399996770105455477, −9.38654661248258730560427134308, −8.40198990478682474273595383671, −7.195532890569075366301781775213, −6.46356520789533275187944360797, −5.17772575736436430224363641439, −4.49072622391574939545897631723, −3.59220926302952049358816545557, −2.61405024684156517339925017930, −1.00803088619799472224885324362, 0.32841439789373272251460069234, 1.346346843592443599052228350477, 2.77407192799362910521370786652, 3.18150283941278416015868269515, 5.0182322915580546011660435447, 5.90880188649396856637744431146, 6.33649662600876220218052506662, 7.787159334384561713140367057196, 8.14809006773792958569090606485, 9.23866883109826540533004549844, 10.31083251288218530349205084226, 11.33367736391494739328767709188, 12.03435089932030216830649518438, 13.01547659784536569057579606475, 13.31791013786839266866647009407, 14.51767959276159000641420053702, 15.36219528823246302248356274370, 16.28072911891327087712669289725, 17.11243501983613208100942960075, 18.05785648445624106641544057959, 18.67695465287171585493400188334, 19.28476657573392525431195341740, 20.070217055658505561251438880035, 21.25465879033750973037653271710, 21.8536386069584552410212794953

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