Properties

Label 1-760-760.443-r0-0-0
Degree 11
Conductor 760760
Sign 0.460+0.887i-0.460 + 0.887i
Analytic cond. 3.529423.52942
Root an. cond. 3.529423.52942
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 + 0.766i)3-s + (0.866 − 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.866 + 0.5i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.342 − 0.939i)33-s i·37-s + 39-s + (0.766 + 0.642i)41-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + (0.866 − 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.866 + 0.5i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.342 − 0.939i)33-s i·37-s + 39-s + (0.766 + 0.642i)41-s + ⋯

Functional equation

Λ(s)=(760s/2ΓR(s)L(s)=((0.460+0.887i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(760s/2ΓR(s)L(s)=((0.460+0.887i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 760760    =    235192^{3} \cdot 5 \cdot 19
Sign: 0.460+0.887i-0.460 + 0.887i
Analytic conductor: 3.529423.52942
Root analytic conductor: 3.529423.52942
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ760(443,)\chi_{760} (443, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 760, (0: ), 0.460+0.887i)(1,\ 760,\ (0:\ ),\ -0.460 + 0.887i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.4134321972+0.6801680654i0.4134321972 + 0.6801680654i
L(12)L(\frac12) \approx 0.4134321972+0.6801680654i0.4134321972 + 0.6801680654i
L(1)L(1) \approx 0.7576383149+0.2568738252i0.7576383149 + 0.2568738252i
L(1)L(1) \approx 0.7576383149+0.2568738252i0.7576383149 + 0.2568738252i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
19 1 1
good3 1+(0.642+0.766i)T 1 + (-0.642 + 0.766i)T
7 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
11 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1+(0.6420.766i)T 1 + (-0.642 - 0.766i)T
17 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
23 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
29 1+(0.173+0.984i)T 1 + (0.173 + 0.984i)T
31 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
37 1iT 1 - iT
41 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
43 1+(0.3420.939i)T 1 + (-0.342 - 0.939i)T
47 1+(0.984+0.173i)T 1 + (-0.984 + 0.173i)T
53 1+(0.342+0.939i)T 1 + (-0.342 + 0.939i)T
59 1+(0.173+0.984i)T 1 + (-0.173 + 0.984i)T
61 1+(0.939+0.342i)T 1 + (0.939 + 0.342i)T
67 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
71 1+(0.9390.342i)T 1 + (0.939 - 0.342i)T
73 1+(0.642+0.766i)T 1 + (-0.642 + 0.766i)T
79 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
83 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
89 1+(0.766+0.642i)T 1 + (-0.766 + 0.642i)T
97 1+(0.9840.173i)T 1 + (-0.984 - 0.173i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−22.11575928865536331248729699458, −21.50296932871845535942479973730, −20.643134015466349221856039416012, −19.40288507219837781355952870230, −18.925434423561648488224372732678, −18.028876946500708767150003255516, −17.48040863199519879441210894146, −16.565093522072225380269111532106, −15.79054768439123103652835125394, −14.65594306525382571132853564046, −13.90532916058386091546218235635, −13.04246028445459096383920598733, −12.19177652388555389920768253825, −11.33869627622453243329802513084, −10.938767328691461962834889337511, −9.63986485877194683240462844117, −8.415452883921464585989392769229, −7.94155362370984090045339831714, −6.77430553199828026753483071637, −6.0227855306820711035423358429, −5.09383655291551403442024802420, −4.27306089886836964473669721795, −2.510839547839773874658873538707, −1.93814389626036392091066916048, −0.422841236686348237670871056774, 1.25740245021139799766955510478, 2.63899648986183781940425559505, 3.879174047194732012323292715924, 4.88000353738630575968331980856, 5.205899717535006215371456801863, 6.58153631464902577322243342097, 7.4564217475766663131675173838, 8.43208066452110896280895559490, 9.5720584549426314261438181822, 10.30714209828863551180844981593, 10.9682341508132892374608389629, 11.83730271295928907469290818967, 12.66127979603714317214080918324, 13.73100513448362566474021602362, 14.77404004103721040189053146496, 15.33867572806124537779954877865, 16.1179920850621920738490634257, 17.21683597966165589515765773194, 17.66971060457041640929557639546, 18.241082294841189741087177253212, 19.85275461245167187826454502605, 20.2589131499033481853837983450, 21.13776478749200477790896468250, 21.86400520391079985495350722780, 22.648150593608323453870831503604

Graph of the ZZ-function along the critical line