Properties

Label 1-520-520.323-r1-0-0
Degree 11
Conductor 520520
Sign 0.492+0.870i0.492 + 0.870i
Analytic cond. 55.881755.8817
Root an. cond. 55.881755.8817
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  + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s i·21-s + (−0.866 − 0.5i)23-s i·27-s + (−0.5 + 0.866i)29-s i·31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.866 − 0.5i)41-s + (−0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)7-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s i·21-s + (−0.866 − 0.5i)23-s i·27-s + (−0.5 + 0.866i)29-s i·31-s + (−0.5 − 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.866 − 0.5i)41-s + (−0.866 + 0.5i)43-s + ⋯

Functional equation

Λ(s)=(520s/2ΓR(s+1)L(s)=((0.492+0.870i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(520s/2ΓR(s+1)L(s)=((0.492+0.870i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 520520    =    235132^{3} \cdot 5 \cdot 13
Sign: 0.492+0.870i0.492 + 0.870i
Analytic conductor: 55.881755.8817
Root analytic conductor: 55.881755.8817
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ520(323,)\chi_{520} (323, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 520, (1: ), 0.492+0.870i)(1,\ 520,\ (1:\ ),\ 0.492 + 0.870i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.6636449158+0.3869869330i0.6636449158 + 0.3869869330i
L(12)L(\frac12) \approx 0.6636449158+0.3869869330i0.6636449158 + 0.3869869330i
L(1)L(1) \approx 0.72659866750.09458026511i0.7265986675 - 0.09458026511i
L(1)L(1) \approx 0.72659866750.09458026511i0.7265986675 - 0.09458026511i

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
13 1 1
good3 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
7 1+(0.50.866i)T 1 + (-0.5 - 0.866i)T
11 1+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
17 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
19 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
23 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
29 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
31 1iT 1 - iT
37 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
41 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
43 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
47 1+T 1 + T
53 1iT 1 - iT
59 1+(0.8660.5i)T 1 + (0.866 - 0.5i)T
61 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
67 1+(0.5+0.866i)T 1 + (-0.5 + 0.866i)T
71 1+(0.866+0.5i)T 1 + (-0.866 + 0.5i)T
73 1+T 1 + T
79 1+T 1 + T
83 1T 1 - T
89 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
97 1+(0.50.866i)T 1 + (-0.5 - 0.866i)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

−23.15711381502983419708255455797, −22.11737363735985194235170948289, −21.77038412066868538425815958928, −20.96837051509638667997825940993, −19.671309248951886153925442658234, −18.96407320836640349049199447535, −18.03567472329474650419818560488, −17.0947739586967297097193874063, −16.48304423275552298497319113981, −15.541865284507681905987115509832, −14.90769262395361645855257501148, −13.73652980739283025671428808389, −12.48501047125533556928936882207, −11.99767783764473353296571479223, −11.072739204705681359621597088345, −10.09934976977758331189909878587, −9.27884882784566253948609640892, −8.40476055823002741646050051855, −6.90532112515624693582180064726, −6.05407138218252033419135812345, −5.43241337313868144820743911, −4.14257312719607051742372092721, −3.289446203643195813809376519010, −1.73580786022444127632088196002, −0.274160252048138972248955797836, 0.90744489531020905139632983594, 1.97859557052017426063146354644, 3.62060200655966760818896619328, 4.52933397187883971285121491851, 5.71223492922249817082316495336, 6.65679819038091384567082706367, 7.25584623265321186630208274876, 8.35052044936438626950448190159, 9.81438168908252408654724181413, 10.33862946528627349338529800283, 11.4678172456547338710734292497, 12.22523161349163296482410573376, 13.012631114290699640693886309129, 13.92678374753089733993449440638, 14.83402102991241313510256382639, 16.17906745016025737284686749840, 16.77571824438346819582938383139, 17.36016860381824870351373165067, 18.433395764558228856126122786632, 19.12815303332175407415851657811, 20.05675130473476274673557187150, 20.879208538686423712937482654954, 22.25799008018673869695307785991, 22.52197868080147964763821920714, 23.52599078551723180958146220447

Graph of the ZZ-function along the critical line