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Properties

Label 1-2557-2557.493-r1-0-0
Degree $1$
Conductor $2557$
Sign $0.971 + 0.238i$
Analytic cond. $274.787$
Root an. cond. $274.787$
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  + (0.945 + 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

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

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $0.971 + 0.238i$
Analytic conductor: \(274.787\)
Root analytic conductor: \(274.787\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (1:\ ),\ 0.971 + 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.718230179 + 0.5712674080i\)
\(L(\frac12)\) \(\approx\) \(4.718230179 + 0.5712674080i\)
\(L(1)\) \(\approx\) \(1.970001017 + 0.1172840284i\)
\(L(1)\) \(\approx\) \(1.970001017 + 0.1172840284i\)

Euler product

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

−19.56294029182376129410203644369, −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 $Z$-function along the critical line