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Properties

Label 1-2557-2557.1001-r0-0-0
Degree $1$
Conductor $2557$
Sign $0.260 + 0.965i$
Analytic cond. $11.8746$
Root an. cond. $11.8746$
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  + (−0.947 + 0.318i)2-s + (0.847 + 0.531i)3-s + (0.796 − 0.604i)4-s + (0.943 + 0.332i)5-s + (−0.972 − 0.233i)6-s + (−0.960 − 0.276i)7-s + (−0.562 + 0.826i)8-s + (0.434 + 0.900i)9-s + (−0.999 − 0.0147i)10-s + (0.876 + 0.480i)11-s + (0.996 − 0.0883i)12-s + (0.353 + 0.935i)13-s + (0.999 − 0.0442i)14-s + (0.621 + 0.783i)15-s + (0.269 − 0.963i)16-s + (0.805 − 0.592i)17-s + ⋯
L(s)  = 1  + (−0.947 + 0.318i)2-s + (0.847 + 0.531i)3-s + (0.796 − 0.604i)4-s + (0.943 + 0.332i)5-s + (−0.972 − 0.233i)6-s + (−0.960 − 0.276i)7-s + (−0.562 + 0.826i)8-s + (0.434 + 0.900i)9-s + (−0.999 − 0.0147i)10-s + (0.876 + 0.480i)11-s + (0.996 − 0.0883i)12-s + (0.353 + 0.935i)13-s + (0.999 − 0.0442i)14-s + (0.621 + 0.783i)15-s + (0.269 − 0.963i)16-s + (0.805 − 0.592i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (1001, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546085253 + 1.184037138i\)
\(L(\frac12)\) \(\approx\) \(1.546085253 + 1.184037138i\)
\(L(1)\) \(\approx\) \(1.085965551 + 0.4557660553i\)
\(L(1)\) \(\approx\) \(1.085965551 + 0.4557660553i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (-0.947 + 0.318i)T \)
3 \( 1 + (0.847 + 0.531i)T \)
5 \( 1 + (0.943 + 0.332i)T \)
7 \( 1 + (-0.960 - 0.276i)T \)
11 \( 1 + (0.876 + 0.480i)T \)
13 \( 1 + (0.353 + 0.935i)T \)
17 \( 1 + (0.805 - 0.592i)T \)
19 \( 1 + (0.989 + 0.146i)T \)
23 \( 1 + (-0.394 - 0.918i)T \)
29 \( 1 + (-0.688 - 0.725i)T \)
31 \( 1 + (0.168 - 0.985i)T \)
37 \( 1 + (0.720 + 0.693i)T \)
41 \( 1 + (0.869 - 0.493i)T \)
43 \( 1 + (-0.610 - 0.792i)T \)
47 \( 1 + (-0.353 + 0.935i)T \)
53 \( 1 + (-0.367 + 0.930i)T \)
59 \( 1 + (-0.339 + 0.940i)T \)
61 \( 1 + (0.984 - 0.176i)T \)
67 \( 1 + (-0.778 - 0.627i)T \)
71 \( 1 + (0.890 - 0.454i)T \)
73 \( 1 + (0.586 - 0.809i)T \)
79 \( 1 + (-0.759 + 0.650i)T \)
83 \( 1 + (0.994 + 0.103i)T \)
89 \( 1 + (0.339 - 0.940i)T \)
97 \( 1 + (0.964 - 0.262i)T \)
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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

−19.299466026618316095837427135518, −18.5758275262474780272518687096, −17.920070884143166806923548218956, −17.42900256779181592624905103894, −16.36803543367942731942875484735, −16.04322875149264242940669709906, −14.95616572370942668333639753570, −14.23925664744485987969481754401, −13.28326810923038480031775815134, −12.85177945845245204113875706029, −12.1996125445083795555593284835, −11.30772249449489439431197236334, −10.16842920414176482924776766194, −9.6732087712819335073712706096, −9.156372979287453487528774986625, −8.45986539817824019751118916039, −7.739828726583691001589540188719, −6.81984153904016366693096683678, −6.17088364051014690231596141052, −5.4779193564961846484501372090, −3.562259477223439600405412577475, −3.36491658255377835354433490714, −2.41261636763512944954253173625, −1.43219667387071215335100903399, −0.91944789901868166044670283936, 1.0754516023301491808797170964, 2.006912954130299806552310273128, 2.729153304974566746816727558257, 3.59274162352899428908045141950, 4.59289681241361072415106669308, 5.78996793208095855282908316460, 6.40087321468073400149375352069, 7.18350083873164230398007203578, 7.823758756177087175524991682786, 8.990116537342853878033512526644, 9.49017778173801220258692784149, 9.749423163843909644311432684349, 10.47937576540061630934899705422, 11.397043804922429175005358501058, 12.299696379026905443989960249578, 13.48771538571356815771956961563, 14.0138708181733665673501462826, 14.58706970089193548036882204001, 15.34049294067616233207055118032, 16.24654608102986013117718438800, 16.64234215574047413047620661272, 17.24178286541822559773550398495, 18.43838867249248144511352448078, 18.71215827345587451601533295816, 19.44826914396023234492815368073

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