Properties

Label 1-80-80.19-r1-0-0
Degree $1$
Conductor $80$
Sign $-0.923 - 0.382i$
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  + i·3-s − 7-s − 9-s + i·11-s i·13-s − 17-s i·19-s i·21-s − 23-s i·27-s + i·29-s − 31-s − 33-s + i·37-s + 39-s + ⋯
L(s)  = 1  + i·3-s − 7-s − 9-s + i·11-s i·13-s − 17-s i·19-s i·21-s − 23-s i·27-s + i·29-s − 31-s − 33-s + i·37-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(80\)    =    \(2^{4} \cdot 5\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(8.59719\)
Root analytic conductor: \(8.59719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{80} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 80,\ (1:\ ),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06050106139 + 0.3041593752i\)
\(L(\frac12)\) \(\approx\) \(-0.06050106139 + 0.3041593752i\)
\(L(1)\) \(\approx\) \(0.6490082549 + 0.2688280213i\)
\(L(1)\) \(\approx\) \(0.6490082549 + 0.2688280213i\)

Euler product

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

−30.054765179056206859261430109545, −29.154159444805894128087749276469, −28.51795806731422008873390517175, −26.77439568377164270801753785586, −25.870047229084608614769333256824, −24.73045967566660650225496406032, −23.858054673733082498876048453350, −22.77234370120555460659974227435, −21.68343617014007170731203535082, −20.08266926545199931031108919019, −19.147867432423578633105717059133, −18.38463001490553815179797928561, −16.94377096047829303010143215725, −15.995512051187264359553029930803, −14.23228950840737157581629899140, −13.370924599599317017520944708372, −12.26290315233048464977216954868, −11.112812824637837644876221934105, −9.41590537706542804344715977116, −8.18456517199584090192804828365, −6.74028866790318099628048572604, −5.88981876392024743960749689627, −3.68087883657849355729892612389, −2.07675048939256145975620763136, −0.13853650194408508948009201069, 2.71794826130170972459527086965, 4.09769813699763886205507389319, 5.45881613904210370021931340275, 6.964059069131927889310429754134, 8.74821524100515707299788341949, 9.82206201037914464819291395220, 10.74836471424836936082390488716, 12.29319032108009203323924958783, 13.53685091930915154448259132396, 15.13059340113388784236150144731, 15.72356553163442059323948050815, 16.9777548637102204668459949360, 18.093788426827801775443253356595, 19.91969291626378757328000731117, 20.26258910072587233897931977120, 22.008620328360075262707684951018, 22.4079917877188291524873447196, 23.68598838114093829757165996084, 25.411595155682671673588879220606, 25.97301765456541176812088132358, 27.184866223128984334025766963765, 28.1888168620164840246988983474, 29.01169317334837132170979913394, 30.43401159894429032474277119489, 31.612913252279338373464364516012

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