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Properties

Label 1-980-980.27-r1-0-0
Degree $1$
Conductor $980$
Sign $-0.0406 + 0.999i$
Analytic cond. $105.315$
Root an. cond. $105.315$
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  + (−0.433 + 0.900i)3-s + (−0.623 − 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.781 + 0.623i)13-s + (0.974 + 0.222i)17-s − 19-s + (0.974 − 0.222i)23-s + (0.974 − 0.222i)27-s + (0.222 − 0.974i)29-s + 31-s + (−0.433 − 0.900i)33-s + (−0.974 − 0.222i)37-s + (−0.900 + 0.433i)39-s + (0.900 + 0.433i)41-s + (−0.433 − 0.900i)43-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)3-s + (−0.623 − 0.781i)9-s + (−0.623 + 0.781i)11-s + (0.781 + 0.623i)13-s + (0.974 + 0.222i)17-s − 19-s + (0.974 − 0.222i)23-s + (0.974 − 0.222i)27-s + (0.222 − 0.974i)29-s + 31-s + (−0.433 − 0.900i)33-s + (−0.974 − 0.222i)37-s + (−0.900 + 0.433i)39-s + (0.900 + 0.433i)41-s + (−0.433 − 0.900i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0406 + 0.999i$
Analytic conductor: \(105.315\)
Root analytic conductor: \(105.315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (1:\ ),\ -0.0406 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.159123975 + 1.207240296i\)
\(L(\frac12)\) \(\approx\) \(1.159123975 + 1.207240296i\)
\(L(1)\) \(\approx\) \(0.8997885037 + 0.3472190782i\)
\(L(1)\) \(\approx\) \(0.8997885037 + 0.3472190782i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.433 + 0.900i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.781 + 0.623i)T \)
17 \( 1 + (0.974 + 0.222i)T \)
19 \( 1 - T \)
23 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 + (0.222 - 0.974i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (-0.433 - 0.900i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.974 + 0.222i)T \)
59 \( 1 + (0.900 - 0.433i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 - iT \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.781 + 0.623i)T \)
79 \( 1 + T \)
83 \( 1 + (0.781 - 0.623i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 - iT \)
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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

−21.200171406139525144129572800003, −20.67530438414669990473897354566, −19.3721121537120517613812198805, −19.03424076194535155684766198407, −18.18435372891276265571055792327, −17.517266064979422474262414388376, −16.63428038222482811228911737872, −15.97858689451685696787412453550, −14.92818724276604295056383964227, −13.95226197796856502684946804601, −13.25179547934673517430523099724, −12.62932531210455810186626911194, −11.740682387676257699694880574564, −10.848361052060157630574778364576, −10.34412977928540016666649442367, −8.8347706879171393080274935007, −8.219652728376874371049030610035, −7.38659737703204100699418961456, −6.42784083615888691927148884404, −5.67638425700553172108538522653, −4.93003760071518415008663083230, −3.42647026280304677918893205329, −2.635331993228857995674989194875, −1.328212131556025045891024664105, −0.54015640233406637431191502092, 0.79461577804610326430204039260, 2.20003174064296908813639639845, 3.35589499859838987676936968158, 4.27755933441020530762876583487, 4.98876756915713711912640928673, 5.97842913511178487546986233933, 6.7513949300951230443676163342, 7.97098270043003565365226236358, 8.838734863027853177305343063642, 9.72214124476954125488575511315, 10.44907190281340550153938017662, 11.12198468283038399283999535607, 12.069308836210484763835666897660, 12.80166252273609283689917741972, 13.90446234124619711572906557078, 14.76260864785233079376800695810, 15.49820913231795218929639671161, 16.09568996804914055715367033353, 17.10685452629223016472348749662, 17.47378464493531082319541401591, 18.662857551019778979469299451004, 19.25205823256158888310417978347, 20.56596028964109843838319417900, 20.951421929048514295636926327127, 21.52528236719388405637194385264

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