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Properties

Label 1-2665-2665.1628-r0-0-0
Degree $1$
Conductor $2665$
Sign $0.998 - 0.0464i$
Analytic cond. $12.3762$
Root an. cond. $12.3762$
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  + (−0.669 − 0.743i)2-s + (0.258 + 0.965i)3-s + (−0.104 + 0.994i)4-s + (0.544 − 0.838i)6-s + (0.0523 + 0.998i)7-s + (0.809 − 0.587i)8-s + (−0.866 + 0.5i)9-s + (0.629 − 0.777i)11-s + (−0.987 + 0.156i)12-s + (0.707 − 0.707i)14-s + (−0.978 − 0.207i)16-s + (0.358 + 0.933i)17-s + (0.951 + 0.309i)18-s + (−0.0523 − 0.998i)19-s + (−0.951 + 0.309i)21-s + (−0.998 + 0.0523i)22-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (0.258 + 0.965i)3-s + (−0.104 + 0.994i)4-s + (0.544 − 0.838i)6-s + (0.0523 + 0.998i)7-s + (0.809 − 0.587i)8-s + (−0.866 + 0.5i)9-s + (0.629 − 0.777i)11-s + (−0.987 + 0.156i)12-s + (0.707 − 0.707i)14-s + (−0.978 − 0.207i)16-s + (0.358 + 0.933i)17-s + (0.951 + 0.309i)18-s + (−0.0523 − 0.998i)19-s + (−0.951 + 0.309i)21-s + (−0.998 + 0.0523i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0464i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0464i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2665\)    =    \(5 \cdot 13 \cdot 41\)
Sign: $0.998 - 0.0464i$
Analytic conductor: \(12.3762\)
Root analytic conductor: \(12.3762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2665} (1628, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2665,\ (0:\ ),\ 0.998 - 0.0464i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.318145374 + 0.03060617960i\)
\(L(\frac12)\) \(\approx\) \(1.318145374 + 0.03060617960i\)
\(L(1)\) \(\approx\) \(0.8962728482 + 0.03758600815i\)
\(L(1)\) \(\approx\) \(0.8962728482 + 0.03758600815i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.669 - 0.743i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
7 \( 1 + (0.0523 + 0.998i)T \)
11 \( 1 + (0.629 - 0.777i)T \)
17 \( 1 + (0.358 + 0.933i)T \)
19 \( 1 + (-0.0523 - 0.998i)T \)
23 \( 1 + (0.743 - 0.669i)T \)
29 \( 1 + (0.933 + 0.358i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (-0.978 + 0.207i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.987 - 0.156i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (0.933 + 0.358i)T \)
71 \( 1 + (-0.358 - 0.933i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.544 - 0.838i)T \)
97 \( 1 + (0.777 - 0.629i)T \)
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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

−19.19915034442174539968653634159, −18.56218863365337985596632074449, −17.83437549302323138258274494899, −17.18484106520039809445872982545, −16.84738098441929588539444124586, −15.850527625569791228189100349956, −15.018778684854973427544667677703, −14.26455065242649898920169700026, −13.859841864383537047492188313770, −13.158746867571125442102194977764, −12.0617606607322946294617665298, −11.54944282315656887621667906816, −10.40378248264835555532369243727, −9.86574211667616104142876900953, −9.02859580163664331593854624188, −8.21969290841232643235367798184, −7.57548719207261784119269880867, −6.93054434892232717638479195448, −6.53415864348443056161307985431, −5.47423841276618953065693996783, −4.63499695371933151106123531017, −3.59164714648918824244222744764, −2.45297375456403058500248677508, −1.34330482844495801924343107936, −0.95384702691470271604339592710, 0.65887103042945703805780375412, 1.953113934218132425422553953396, 2.765214892310675514964407858492, 3.37203509922823881923145269983, 4.23696343764625961482479369300, 5.04280075526157333112967090660, 5.99694662378977710518374563473, 6.92546531057948617660625856290, 8.24360548285345293259239507208, 8.5885518376051400471788803008, 9.13598630370813125579231768455, 9.912361628302022424080530145907, 10.663822859836318221694819046311, 11.28230174101401488370097713624, 11.914110557872988856053572350456, 12.6983888355725567983764640188, 13.59145190748532029994777203231, 14.39769261726872279797932024719, 15.22662901323535775784422947329, 15.80820015482257119334957420982, 16.667388499162353100756284453856, 17.11402596025054991455003793422, 17.94567257378547179726167462623, 18.81179596320949304594661356233, 19.42092372749149063007085014496

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