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Properties

Label 1-2057-2057.397-r1-0-0
Degree $1$
Conductor $2057$
Sign $-0.999 + 0.00848i$
Analytic cond. $221.055$
Root an. cond. $221.055$
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  + (−0.877 − 0.479i)2-s + (0.923 − 0.382i)3-s + (0.540 + 0.841i)4-s + (−0.447 + 0.894i)5-s + (−0.994 − 0.106i)6-s + (−0.999 + 0.0356i)7-s + (−0.0713 − 0.997i)8-s + (0.707 − 0.707i)9-s + (0.821 − 0.570i)10-s + (0.821 + 0.570i)12-s + (−0.540 − 0.841i)13-s + (0.894 + 0.447i)14-s + (−0.0713 + 0.997i)15-s + (−0.415 + 0.909i)16-s + (−0.959 + 0.281i)18-s + (0.599 − 0.800i)19-s + ⋯
L(s)  = 1  + (−0.877 − 0.479i)2-s + (0.923 − 0.382i)3-s + (0.540 + 0.841i)4-s + (−0.447 + 0.894i)5-s + (−0.994 − 0.106i)6-s + (−0.999 + 0.0356i)7-s + (−0.0713 − 0.997i)8-s + (0.707 − 0.707i)9-s + (0.821 − 0.570i)10-s + (0.821 + 0.570i)12-s + (−0.540 − 0.841i)13-s + (0.894 + 0.447i)14-s + (−0.0713 + 0.997i)15-s + (−0.415 + 0.909i)16-s + (−0.959 + 0.281i)18-s + (0.599 − 0.800i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2057\)    =    \(11^{2} \cdot 17\)
Sign: $-0.999 + 0.00848i$
Analytic conductor: \(221.055\)
Root analytic conductor: \(221.055\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2057} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2057,\ (1:\ ),\ -0.999 + 0.00848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002993841770 - 0.7052965543i\)
\(L(\frac12)\) \(\approx\) \(0.002993841770 - 0.7052965543i\)
\(L(1)\) \(\approx\) \(0.7273162479 - 0.2493487843i\)
\(L(1)\) \(\approx\) \(0.7273162479 - 0.2493487843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.877 - 0.479i)T \)
3 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (-0.447 + 0.894i)T \)
7 \( 1 + (-0.999 + 0.0356i)T \)
13 \( 1 + (-0.540 - 0.841i)T \)
19 \( 1 + (0.599 - 0.800i)T \)
23 \( 1 + (0.731 - 0.681i)T \)
29 \( 1 + (0.860 - 0.510i)T \)
31 \( 1 + (-0.984 - 0.177i)T \)
37 \( 1 + (0.984 + 0.177i)T \)
41 \( 1 + (0.106 - 0.994i)T \)
43 \( 1 + (0.0713 + 0.997i)T \)
47 \( 1 + (-0.281 + 0.959i)T \)
53 \( 1 + (-0.349 + 0.936i)T \)
59 \( 1 + (0.877 - 0.479i)T \)
61 \( 1 + (-0.627 + 0.778i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.510 - 0.860i)T \)
73 \( 1 + (-0.999 - 0.0356i)T \)
79 \( 1 + (0.315 + 0.948i)T \)
83 \( 1 + (0.349 - 0.936i)T \)
89 \( 1 + (-0.989 - 0.142i)T \)
97 \( 1 + (-0.948 - 0.315i)T \)
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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

−19.97886882498284478704717908651, −19.303113734065118488111398535944, −18.91675957706978362108904838319, −17.944383986750208685411844640528, −16.77032977329998874477945837914, −16.43823592358934174710321413819, −15.923121297045042793987692706475, −15.14307319445939833716058863406, −14.44846113345657006570435385315, −13.61650381203331344123883749274, −12.80373962330147720548860593321, −11.94625517453059598030824023063, −11.03168756120018981536903498289, −9.88954816849226245391964208365, −9.619625665093764803812344214781, −8.8836219543041997123620115490, −8.283342281868693764758577640444, −7.3774053946773849738895966737, −6.885878668847399054088906956276, −5.64602111848544869220632421488, −4.87188095850826914871870289288, −3.88469963535454359678761493462, −3.02217103044897449582472545803, −1.93359476828040644532318074923, −1.05108748252538403393941522097, 0.17341925291050344348075276689, 0.999398648099180103242614073551, 2.4700896896915546781299517849, 2.81121278583415612291369182725, 3.426564811748386887322742938599, 4.39435543668044170042437642897, 6.106564857906284395592830288025, 6.86571104599797198352761359030, 7.43869743083966815655343126341, 8.04483079582749243196629131017, 9.015422651457766931793519729715, 9.62067049170091990622078630117, 10.30103792294205185838805253117, 11.050626203526236532823581647305, 12.004730057331150443986388092450, 12.71230215731913054237362115433, 13.258967766083858025816551892835, 14.27445383592283206554091881407, 15.1515467881007955340366592258, 15.64628003587552873999502682642, 16.397169829403293811589619030578, 17.50614109226007492693300336091, 18.11435555593066537298088066954, 18.80585158167290352910723313579, 19.344987811056794508488096198006

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