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Properties

Label 1-520-520.323-r1-0-0
Degree $1$
Conductor $520$
Sign $0.492 + 0.870i$
Analytic cond. $55.8817$
Root an. cond. $55.8817$
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  + (−0.866 − 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

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

Invariants

Degree: \(1\)
Conductor: \(520\)    =    \(2^{3} \cdot 5 \cdot 13\)
Sign: $0.492 + 0.870i$
Analytic conductor: \(55.8817\)
Root analytic conductor: \(55.8817\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{520} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 520,\ (1:\ ),\ 0.492 + 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6636449158 + 0.3869869330i\)
\(L(\frac12)\) \(\approx\) \(0.6636449158 + 0.3869869330i\)
\(L(1)\) \(\approx\) \(0.7265986675 - 0.09458026511i\)
\(L(1)\) \(\approx\) \(0.7265986675 - 0.09458026511i\)

Euler product

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

−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 $Z$-function along the critical line