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Properties

Label 1-4693-4693.1013-r0-0-0
Degree $1$
Conductor $4693$
Sign $0.926 - 0.375i$
Analytic cond. $21.7942$
Root an. cond. $21.7942$
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  + (0.729 + 0.684i)2-s + (0.515 + 0.856i)3-s + (0.0642 + 0.997i)4-s + (−0.384 + 0.922i)5-s + (−0.209 + 0.977i)6-s + (−0.993 + 0.110i)7-s + (−0.635 + 0.771i)8-s + (−0.467 + 0.883i)9-s + (−0.912 + 0.410i)10-s + (−0.137 − 0.990i)11-s + (−0.821 + 0.569i)12-s + (−0.800 − 0.599i)14-s + (−0.989 + 0.146i)15-s + (−0.991 + 0.128i)16-s + (0.832 + 0.554i)17-s + (−0.945 + 0.324i)18-s + ⋯
L(s)  = 1  + (0.729 + 0.684i)2-s + (0.515 + 0.856i)3-s + (0.0642 + 0.997i)4-s + (−0.384 + 0.922i)5-s + (−0.209 + 0.977i)6-s + (−0.993 + 0.110i)7-s + (−0.635 + 0.771i)8-s + (−0.467 + 0.883i)9-s + (−0.912 + 0.410i)10-s + (−0.137 − 0.990i)11-s + (−0.821 + 0.569i)12-s + (−0.800 − 0.599i)14-s + (−0.989 + 0.146i)15-s + (−0.991 + 0.128i)16-s + (0.832 + 0.554i)17-s + (−0.945 + 0.324i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4693 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4693\)    =    \(13 \cdot 19^{2}\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(21.7942\)
Root analytic conductor: \(21.7942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4693} (1013, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4693,\ (0:\ ),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3238081640 + 0.06308816358i\)
\(L(\frac12)\) \(\approx\) \(-0.3238081640 + 0.06308816358i\)
\(L(1)\) \(\approx\) \(0.6390179656 + 0.9670934216i\)
\(L(1)\) \(\approx\) \(0.6390179656 + 0.9670934216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.729 + 0.684i)T \)
3 \( 1 + (0.515 + 0.856i)T \)
5 \( 1 + (-0.384 + 0.922i)T \)
7 \( 1 + (-0.993 + 0.110i)T \)
11 \( 1 + (-0.137 - 0.990i)T \)
17 \( 1 + (0.832 + 0.554i)T \)
23 \( 1 + (-0.999 - 0.0183i)T \)
29 \( 1 + (-0.971 - 0.236i)T \)
31 \( 1 + (-0.451 + 0.892i)T \)
37 \( 1 + (-0.789 + 0.614i)T \)
41 \( 1 + (0.979 + 0.200i)T \)
43 \( 1 + (0.100 + 0.994i)T \)
47 \( 1 + (-0.741 - 0.670i)T \)
53 \( 1 + (-0.951 + 0.307i)T \)
59 \( 1 + (-0.315 - 0.948i)T \)
61 \( 1 + (0.983 + 0.182i)T \)
67 \( 1 + (0.00918 + 0.999i)T \)
71 \( 1 + (-0.983 + 0.182i)T \)
73 \( 1 + (0.896 - 0.443i)T \)
79 \( 1 + (0.811 - 0.584i)T \)
83 \( 1 + (0.298 - 0.954i)T \)
89 \( 1 + (0.896 + 0.443i)T \)
97 \( 1 + (0.00918 - 0.999i)T \)
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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

−17.74660695740184949669823853732, −16.818747020333365128628441932025, −16.02727920695847804409443282951, −15.4517270739117065615602865272, −14.667986481596531635298699605546, −13.9980596841331316699871983664, −13.302772497789276600437943175575, −12.7471558763443629962324718452, −12.33515895860218711137515493455, −11.87790365585736346312116126161, −10.94516145279248886334974685099, −9.81762795185606353125957556681, −9.51586440404527775672443498946, −8.82442385208720996664166580801, −7.66547688211891563071399728252, −7.30906245229498953942231340191, −6.30869976487172969711762693710, −5.6619319734825121162241012709, −4.88852564827022697531090491093, −3.81103987165899095264315275252, −3.58345421988612418696950212270, −2.44728319181293304045039822618, −1.88877437801733119558018395427, −0.95246530933127372212791542011, −0.06606215316071778540360440314, 2.07251484752438490975817194909, 3.00877180116080715024545126765, 3.43089283142091137518182320157, 3.81298214498708941985720430759, 4.810020448554437587309393896646, 5.81036498355748469680505498562, 6.11416493726311781077277538999, 7.04892742514543070667895749502, 7.859182401056971499511976856936, 8.32579081777738615810761578292, 9.202957279781716573682949104579, 9.967734919413200888247553538967, 10.67168612519340754673427893885, 11.369313561898907095365964684842, 12.1125658185296350000268768258, 13.0022815853357333609640477844, 13.60891492599712464588048472239, 14.42596571449216688965113448312, 14.61221099994706039411335573389, 15.52860876706371054917077217038, 16.03184973379338813175677613414, 16.37314622211878594985481914988, 17.15646561444780788550530504376, 18.11815899365149891735302685785, 18.96593256055970273444380214392

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