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Properties

Label 1-2205-2205.34-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.928 - 0.371i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)8-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s − 19-s + (0.988 − 0.149i)22-s + (0.988 − 0.149i)23-s + (−0.623 − 0.781i)26-s + (−0.988 − 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.365 − 0.930i)32-s + (−0.826 − 0.563i)34-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)2-s + (−0.988 + 0.149i)4-s + (0.222 + 0.974i)8-s + (0.0747 + 0.997i)11-s + (0.826 − 0.563i)13-s + (0.955 − 0.294i)16-s + (0.623 − 0.781i)17-s − 19-s + (0.988 − 0.149i)22-s + (0.988 − 0.149i)23-s + (−0.623 − 0.781i)26-s + (−0.988 − 0.149i)29-s + (0.5 − 0.866i)31-s + (−0.365 − 0.930i)32-s + (−0.826 − 0.563i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.928 - 0.371i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.928 - 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2384164830 - 1.235945334i\)
\(L(\frac12)\) \(\approx\) \(0.2384164830 - 1.235945334i\)
\(L(1)\) \(\approx\) \(0.7981671010 - 0.4452834586i\)
\(L(1)\) \(\approx\) \(0.7981671010 - 0.4452834586i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (0.826 - 0.563i)T \)
17 \( 1 + (0.623 - 0.781i)T \)
19 \( 1 - T \)
23 \( 1 + (0.988 - 0.149i)T \)
29 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.623 + 0.781i)T \)
41 \( 1 + (-0.955 - 0.294i)T \)
43 \( 1 + (-0.955 + 0.294i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.733 + 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.826 + 0.563i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−19.52370836663429196855103206999, −18.903827004788844837558756369328, −18.56109993806740903579030776155, −17.45288240620512599451210587414, −16.910903963808302954171083497781, −16.33217490762874007905421672105, −15.59537721216280054411771807010, −14.82039590233678495342377599112, −14.23444331792835403848108661701, −13.39268391209564177533980070192, −12.9317105101313401241690664605, −11.88480977614112409247453074303, −10.901532881096913297494460604167, −10.302586344742083750464218280970, −9.211485917455686882836106327806, −8.59984133228853924490556537232, −8.13206958991911492975185506698, −6.99748643612639361327644052726, −6.44603678133317657366939992866, −5.666380779260814504802934805200, −4.95618367329434281531544855106, −3.81744927041878061166385704279, −3.39924204119816389351506368724, −1.78984810156842547243726174626, −0.84839031008214310874102656093, 0.28131604762868941100898478893, 1.272763962593568186866900481257, 2.12191792139601960007503629107, 3.03845852984169348134054703342, 3.80286972871411524071553078784, 4.70652180884477134564577465308, 5.35558850629868970355910129911, 6.44596226121277294169726983036, 7.44392189486165096575381550127, 8.27260275192204759900952754853, 8.983134194281883986054639700720, 9.8550459121954062939507905736, 10.32827392221074787467325750052, 11.28394301490081445320498828888, 11.76569554548622552397669024991, 12.830526493309074315797701827810, 13.05713010013369817606100066832, 14.00818700517475978274017508888, 14.85995642695533077395973151624, 15.42535358679753945313217787994, 16.61692808803909310147561782462, 17.25146704349145936380965810753, 17.91120105175109695376388002819, 18.7606359384750446406390558654, 19.081695688948586120628226766005

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