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Properties

Label 1-1805-1805.8-r0-0-0
Degree $1$
Conductor $1805$
Sign $-0.343 - 0.939i$
Analytic cond. $8.38238$
Root an. cond. $8.38238$
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  + (0.0550 − 0.998i)2-s + (−0.981 + 0.191i)3-s + (−0.993 − 0.110i)4-s + (0.137 + 0.990i)6-s + (0.915 + 0.401i)7-s + (−0.164 + 0.986i)8-s + (0.926 − 0.376i)9-s + (0.789 − 0.614i)11-s + (0.996 − 0.0825i)12-s + (0.656 + 0.754i)13-s + (0.451 − 0.892i)14-s + (0.975 + 0.218i)16-s + (−0.110 − 0.993i)17-s + (−0.324 − 0.945i)18-s + (−0.975 − 0.218i)21-s + (−0.569 − 0.821i)22-s + ⋯
L(s)  = 1  + (0.0550 − 0.998i)2-s + (−0.981 + 0.191i)3-s + (−0.993 − 0.110i)4-s + (0.137 + 0.990i)6-s + (0.915 + 0.401i)7-s + (−0.164 + 0.986i)8-s + (0.926 − 0.376i)9-s + (0.789 − 0.614i)11-s + (0.996 − 0.0825i)12-s + (0.656 + 0.754i)13-s + (0.451 − 0.892i)14-s + (0.975 + 0.218i)16-s + (−0.110 − 0.993i)17-s + (−0.324 − 0.945i)18-s + (−0.975 − 0.218i)21-s + (−0.569 − 0.821i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.343 - 0.939i$
Analytic conductor: \(8.38238\)
Root analytic conductor: \(8.38238\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1805,\ (0:\ ),\ -0.343 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6557360857 - 0.9379355476i\)
\(L(\frac12)\) \(\approx\) \(0.6557360857 - 0.9379355476i\)
\(L(1)\) \(\approx\) \(0.7555223486 - 0.4228430346i\)
\(L(1)\) \(\approx\) \(0.7555223486 - 0.4228430346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0550 - 0.998i)T \)
3 \( 1 + (-0.981 + 0.191i)T \)
7 \( 1 + (0.915 + 0.401i)T \)
11 \( 1 + (0.789 - 0.614i)T \)
13 \( 1 + (0.656 + 0.754i)T \)
17 \( 1 + (-0.110 - 0.993i)T \)
23 \( 1 + (-0.981 - 0.191i)T \)
29 \( 1 + (-0.592 - 0.805i)T \)
31 \( 1 + (-0.546 - 0.837i)T \)
37 \( 1 + (0.614 + 0.789i)T \)
41 \( 1 + (-0.851 - 0.523i)T \)
43 \( 1 + (-0.272 - 0.962i)T \)
47 \( 1 + (0.990 - 0.137i)T \)
53 \( 1 + (0.990 - 0.137i)T \)
59 \( 1 + (0.851 + 0.523i)T \)
61 \( 1 + (0.350 - 0.936i)T \)
67 \( 1 + (0.771 + 0.635i)T \)
71 \( 1 + (-0.350 - 0.936i)T \)
73 \( 1 + (0.110 + 0.993i)T \)
79 \( 1 + (-0.962 + 0.272i)T \)
83 \( 1 + (0.735 - 0.677i)T \)
89 \( 1 + (0.993 + 0.110i)T \)
97 \( 1 + (-0.771 + 0.635i)T \)
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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

−20.39227279796554461036994225931, −19.595790643173318490080924914376, −18.494781246885458720863929480091, −17.81766723575897855902103051885, −17.610427628471536727767452950851, −16.718654702158997485456444230059, −16.20611752029057580179254339821, −15.20623334287990923383954988712, −14.69495825450118857287265169715, −13.824006983372052748377381462968, −12.97138333413312365046199573024, −12.38444445396656459296047860843, −11.45297777283797930048926601397, −10.61666759545516547321686949414, −9.96313291336469772734245222824, −8.85084848293703832097726296501, −8.04317230463903977370520021906, −7.346281955790728782010227580626, −6.612219976951428972004456385242, −5.80689413836525906770358379871, −5.191296195353990847586297114190, −4.249329285961033655474717415110, −3.73191219579824712355736702453, −1.756869142973898946606531623828, −1.01054237754690180533728516724, 0.5862204531860657607248085932, 1.55695594394555923557432203455, 2.35675824365598030863319522495, 3.84230051951137842338924510215, 4.183629791374756612397051484425, 5.24832794403564680252682762070, 5.798213306794660260556702163155, 6.7825249673833506961424921273, 7.97526827894864886063656188644, 8.880429445887396608231240780322, 9.49403442996369679848968709142, 10.411574797379881345066937484499, 11.22117228002483286786998292952, 11.781008272245934237542441753280, 11.92896947499041326524434848414, 13.23706074871585758123811172602, 13.84056391080301746440106366492, 14.64676101926417183411224757493, 15.55087618390417798219550985497, 16.54232514243692052909848943072, 17.13091910183567622790564129104, 17.90972573594841732997506047385, 18.69993830553958110952519631026, 18.83798045717382849552999052863, 20.31199048405048861387089197003

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