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Properties

Label 1-80-80.37-r1-0-0
Degree $1$
Conductor $80$
Sign $-0.160 + 0.987i$
Analytic cond. $8.59719$
Root an. cond. $8.59719$
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  − 3-s − i·7-s + 9-s + i·11-s − 13-s + i·17-s + i·19-s + i·21-s + i·23-s − 27-s + i·29-s + 31-s − i·33-s − 37-s + 39-s + ⋯
L(s)  = 1  − 3-s − i·7-s + 9-s + i·11-s − 13-s + i·17-s + i·19-s + i·21-s + i·23-s − 27-s + i·29-s + 31-s − i·33-s − 37-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(8.59719\)
Root analytic conductor: \(8.59719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (1:\ ),\ -0.160 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4580876265 + 0.5384174684i\)
\(L(\frac12)\) \(\approx\) \(0.4580876265 + 0.5384174684i\)
\(L(1)\) \(\approx\) \(0.6934106513 + 0.1125250580i\)
\(L(1)\) \(\approx\) \(0.6934106513 + 0.1125250580i\)

Euler product

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

−30.32164370224955620771118292660, −29.26101826600404340053590499318, −28.49193103861731306114364831857, −27.44022707048651415509319030672, −26.48321906423382459676124890368, −24.72448557667522278220019273648, −24.26082085392113354386275318053, −22.743076975137250930728938631534, −21.999659258039620981691211433402, −21.064531362734298516302283739960, −19.29972724533990080861284119443, −18.39776936689011162908877433111, −17.28988426184785329570208324117, −16.180763022646995156080745146863, −15.19196298768307426273132952344, −13.57766066601001933426154646987, −12.185160019345464716688592633579, −11.473129799310302997805335433826, −10.08580404606535123808458411804, −8.73376740371387756751185990256, −7.03206798346526007316108109614, −5.77636342756449621079992583198, −4.721517479563469827346336276805, −2.604969463301368035305177692089, −0.39773952566724138125879098491, 1.516296359398396665932960412607, 3.96811598994924732295270336187, 5.1655546174036904767295869785, 6.696144987501418286911809537915, 7.685699204209441400031054861915, 9.82841564546632115817647740351, 10.567707924974459418428867633039, 11.99495242907030483006179611047, 12.90708437388974344177057401085, 14.43172109574250655936967431815, 15.7542327543663487616594482122, 17.10869168709573795423226378923, 17.49497261984707440539595212957, 19.035725758231040018859613264445, 20.24124077496028915504785398303, 21.49607023776683493837428376120, 22.65074056464417705197275761569, 23.428211202222137088561915214522, 24.40057729378688732459795356108, 25.84214186383906217361094020137, 27.054699105000061744893476841048, 27.87859491490267989515616245940, 29.098039569493542142629670752070, 29.79477989456283015647533788403, 30.91602929437331177510937946100

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