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Properties

Label 1-2557-2557.493-r1-0-0
Degree 11
Conductor 25572557
Sign 0.971+0.238i0.971 + 0.238i
Analytic cond. 274.787274.787
Root an. cond. 274.787274.787
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.945 + 0.325i)2-s + (−0.283 − 0.958i)3-s + (0.787 + 0.616i)4-s + (0.132 + 0.991i)5-s + (0.0442 − 0.999i)6-s + (0.598 − 0.801i)7-s + (0.544 + 0.839i)8-s + (−0.839 + 0.544i)9-s + (−0.197 + 0.980i)10-s + (−0.883 − 0.467i)11-s + (0.367 − 0.930i)12-s + (0.814 + 0.580i)13-s + (0.826 − 0.562i)14-s + (0.912 − 0.408i)15-s + (0.240 + 0.970i)16-s + (0.650 + 0.759i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.325i)2-s + (−0.283 − 0.958i)3-s + (0.787 + 0.616i)4-s + (0.132 + 0.991i)5-s + (0.0442 − 0.999i)6-s + (0.598 − 0.801i)7-s + (0.544 + 0.839i)8-s + (−0.839 + 0.544i)9-s + (−0.197 + 0.980i)10-s + (−0.883 − 0.467i)11-s + (0.367 − 0.930i)12-s + (0.814 + 0.580i)13-s + (0.826 − 0.562i)14-s + (0.912 − 0.408i)15-s + (0.240 + 0.970i)16-s + (0.650 + 0.759i)17-s + ⋯

Functional equation

Λ(s)=(2557s/2 ΓR(s+1) L(s)=((0.971+0.238i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2557s/2 ΓR(s+1) L(s)=((0.971+0.238i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 25572557
Sign: 0.971+0.238i0.971 + 0.238i
Analytic conductor: 274.787274.787
Root analytic conductor: 274.787274.787
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2557(493,⋅)\chi_{2557} (493, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2557, (1: ), 0.971+0.238i)(1,\ 2557,\ (1:\ ),\ 0.971 + 0.238i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 4.718230179+0.5712674080i4.718230179 + 0.5712674080i
L(12)L(\frac12) ≈\approx 4.718230179+0.5712674080i4.718230179 + 0.5712674080i
L(1)L(1) ≈\approx 1.970001017+0.1172840284i1.970001017 + 0.1172840284i
L(1)L(1) ≈\approx 1.970001017+0.1172840284i1.970001017 + 0.1172840284i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2557 1 1
good2 1+(0.945+0.325i)T 1 + (0.945 + 0.325i)T
3 1+(−0.283−0.958i)T 1 + (-0.283 - 0.958i)T
5 1+(0.132+0.991i)T 1 + (0.132 + 0.991i)T
7 1+(0.598−0.801i)T 1 + (0.598 - 0.801i)T
11 1+(−0.883−0.467i)T 1 + (-0.883 - 0.467i)T
13 1+(0.814+0.580i)T 1 + (0.814 + 0.580i)T
17 1+(0.650+0.759i)T 1 + (0.650 + 0.759i)T
19 1+(0.408−0.912i)T 1 + (0.408 - 0.912i)T
23 1+(0.0221−0.999i)T 1 + (0.0221 - 0.999i)T
29 1+(0.999−0.0442i)T 1 + (0.999 - 0.0442i)T
31 1+(−0.997+0.0663i)T 1 + (-0.997 + 0.0663i)T
37 1+(0.598+0.801i)T 1 + (0.598 + 0.801i)T
41 1+(0.773−0.633i)T 1 + (0.773 - 0.633i)T
43 1+(−0.219−0.975i)T 1 + (-0.219 - 0.975i)T
47 1+(0.580+0.814i)T 1 + (0.580 + 0.814i)T
53 1+(−0.408−0.912i)T 1 + (-0.408 - 0.912i)T
59 1+(0.730+0.683i)T 1 + (0.730 + 0.683i)T
61 1+(−0.730−0.683i)T 1 + (-0.730 - 0.683i)T
67 1+(0.262+0.964i)T 1 + (0.262 + 0.964i)T
71 1+(0.996−0.0883i)T 1 + (0.996 - 0.0883i)T
73 1+(−0.984+0.176i)T 1 + (-0.984 + 0.176i)T
79 1+(0.132+0.991i)T 1 + (0.132 + 0.991i)T
83 1+(−0.176−0.984i)T 1 + (-0.176 - 0.984i)T
89 1+(0.683−0.730i)T 1 + (0.683 - 0.730i)T
97 1+(−0.903+0.428i)T 1 + (-0.903 + 0.428i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.56294029182376129410203644369, −18.28955589021316376400310219817, −17.924412162528285360095753877293, −16.75540245085411171488119973752, −16.03137014160405749264275287317, −15.73963882113579484140859888638, −14.97220163136789959858732547616, −14.251489011764015929917007776440, −13.46307821653934453198211351078, −12.57846889408860137651979389123, −12.105912812704921871348454664803, −11.36094377375879094027639737987, −10.69268853359572953958918527412, −9.79497500308765846714368481366, −9.310327199627664682630234602332, −8.22111288264281949827679198006, −7.57559215268774591242549549972, −5.98846449515467748530390169922, −5.599235911215280836618329868949, −5.07238210489361929192951771432, −4.40711130818921334295114597116, −3.48338327721296139620402704707, −2.70393427600069184979657934668, −1.64167101076360551392999261204, −0.725024524153775030620263648493, 0.76915448858605996596407617387, 1.8209929545045581150044339285, 2.627519108634435211852822473084, 3.38395251510428733487798614406, 4.327107915765124988871967231787, 5.30633169888534563455695188398, 6.01627661802996747413526945174, 6.704315042397323766946445757351, 7.27780655606681947540537123538, 7.96749112713045302112931275488, 8.60570521518253090973064187894, 10.30885404807166941632675633865, 10.921056844597391882508526516620, 11.28654211086542010885811668410, 12.15823543280316140195515323966, 13.07275562931914317862753822859, 13.563293120534056405944735295329, 14.20776870667217844124120701621, 14.6073166307635122026800492655, 15.67067501337644290519149946037, 16.3775317855770313126659065622, 17.16874278670034343234595304449, 17.79053195441424600064802778079, 18.50853486710979915953513654941, 19.14787026448121019787750561358

Graph of the ZZ-function along the critical line