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Properties

Label 1-104-104.75-r1-0-0
Degree $1$
Conductor $104$
Sign $0.859 - 0.511i$
Analytic cond. $11.1763$
Root an. cond. $11.1763$
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  + (−0.5 + 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + 21-s + (0.5 − 0.866i)23-s + 25-s + 27-s + (0.5 − 0.866i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(11.1763\)
Root analytic conductor: \(11.1763\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 104,\ (1:\ ),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.469759334 - 0.4039388098i\)
\(L(\frac12)\) \(\approx\) \(1.469759334 - 0.4039388098i\)
\(L(1)\) \(\approx\) \(1.067058220 + 0.01368435880i\)
\(L(1)\) \(\approx\) \(1.067058220 + 0.01368435880i\)

Euler product

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

−29.546377427231459564965984488850, −28.514963339777991679736811712851, −28.1069934813781177979727850002, −26.19511223335934151386444880409, −25.22414246240529542316063946072, −24.68443418638207155002417063559, −23.3700121920613906675248915691, −22.2571868728314383588815670824, −21.60365818740086743052357841656, −19.96417260444996972064231645046, −18.98312366500148433006473683608, −17.76952127247303782544422490971, −17.35494160198535760198444363604, −15.85080866286412862827650298906, −14.46904206086167308699540872693, −13.20196219232550764039629956206, −12.51332070105726470030576376978, −11.285543531033590032713819872553, −9.82831371350558415696540602739, −8.72612109800542187876624580358, −7.00974469004392563646352406122, −6.14817623158702127844767004656, −5.02861742291624996869061426539, −2.66866313440044051002788680370, −1.46424222172001278964961693612, 0.76826862225171870991262097020, 3.04594545321128796953490673115, 4.43254381036835566955231921860, 5.79630429585982221964808173847, 6.75799398717826870672030393544, 8.79952005128423266839034177917, 9.88069692202636063957018295517, 10.63933945616126462579863663786, 11.932121411986731881396440339210, 13.5237836784220866875259898522, 14.28269651330522567465458281918, 15.84608787519206771703447491607, 16.76479999640116125118336249946, 17.44727974473013530611299999523, 18.831493384114091035277031247721, 20.361380452743022436234511944017, 21.06441087542475272451805819903, 22.28369218002022411082905718136, 22.80429082056762554636361240684, 24.28042525776768898978638648477, 25.40354549531544962266508782535, 26.63580938991335027383718744034, 27.07977883387228510840204991317, 28.6644305548210518559220803081, 29.188130169917287374168489058598

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