↑

Properties

Label 1-2557-2557.1024-r0-0-0
Degree $1$
Conductor $2557$
Sign $0.647 + 0.761i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.999 + 0.0245i)2-s + (−0.376 + 0.926i)3-s + (0.998 + 0.0491i)4-s + (−0.163 + 0.986i)5-s + (−0.399 + 0.916i)6-s + (0.997 − 0.0687i)7-s + (0.997 + 0.0736i)8-s + (−0.716 − 0.697i)9-s + (−0.188 + 0.982i)10-s + (0.625 − 0.780i)11-s + (−0.421 + 0.906i)12-s + (−0.652 − 0.758i)13-s + (0.999 − 0.0442i)14-s + (−0.852 − 0.523i)15-s + (0.995 + 0.0981i)16-s + (0.723 + 0.690i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0245i)2-s + (−0.376 + 0.926i)3-s + (0.998 + 0.0491i)4-s + (−0.163 + 0.986i)5-s + (−0.399 + 0.916i)6-s + (0.997 − 0.0687i)7-s + (0.997 + 0.0736i)8-s + (−0.716 − 0.697i)9-s + (−0.188 + 0.982i)10-s + (0.625 − 0.780i)11-s + (−0.421 + 0.906i)12-s + (−0.652 − 0.758i)13-s + (0.999 − 0.0442i)14-s + (−0.852 − 0.523i)15-s + (0.995 + 0.0981i)16-s + (0.723 + 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $0.647 + 0.761i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ 0.647 + 0.761i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.168126170 + 1.465141377i\)
\(L(\frac12)\) \(\approx\) \(3.168126170 + 1.465141377i\)
\(L(1)\) \(\approx\) \(1.907838869 + 0.6494351277i\)
\(L(1)\) \(\approx\) \(1.907838869 + 0.6494351277i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.999 + 0.0245i)T \)
3 \( 1 + (-0.376 + 0.926i)T \)
5 \( 1 + (-0.163 + 0.986i)T \)
7 \( 1 + (0.997 - 0.0687i)T \)
11 \( 1 + (0.625 - 0.780i)T \)
13 \( 1 + (-0.652 - 0.758i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (0.316 - 0.948i)T \)
23 \( 1 + (0.288 - 0.957i)T \)
29 \( 1 + (-0.833 + 0.552i)T \)
31 \( 1 + (-0.941 - 0.337i)T \)
37 \( 1 + (0.808 - 0.588i)T \)
41 \( 1 + (0.983 + 0.180i)T \)
43 \( 1 + (0.302 + 0.953i)T \)
47 \( 1 + (0.652 - 0.758i)T \)
53 \( 1 + (-0.367 + 0.930i)T \)
59 \( 1 + (0.867 + 0.497i)T \)
61 \( 1 + (0.984 - 0.176i)T \)
67 \( 1 + (0.946 + 0.323i)T \)
71 \( 1 + (-0.681 + 0.731i)T \)
73 \( 1 + (-0.0712 - 0.997i)T \)
79 \( 1 + (-0.759 + 0.650i)T \)
83 \( 1 + (-0.969 + 0.243i)T \)
89 \( 1 + (0.864 - 0.502i)T \)
97 \( 1 + (-0.709 - 0.704i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.35154331497472180770516217705, −18.79225694761908964327421085514, −17.66128328678139289116513205010, −17.107229096669617077523000199609, −16.55872990642417657057141591017, −15.78375678495109277995903705277, −14.58741515468172223890826495089, −14.393654010243148278308518830645, −13.52340674181893381372043821886, −12.73247145530824249131491392783, −12.16209979331469440201833765787, −11.66478402539764162173989381700, −11.22667205306990379718762499290, −9.910700306706066731120036951288, −9.08719298042319109130691572248, −7.86104832509135416349665707284, −7.55981985255908429547031741368, −6.81471174608863476899993015025, −5.59670394233836366994343627153, −5.34453358660579760247255509062, −4.47999805200553908344532921688, −3.759294272377352677313578589209, −2.35395791094079317968316788864, −1.690296351590607385631773326908, −1.10734199031522568235584043016, 0.96398893087729547446126398731, 2.38996243698242909034504092504, 3.05864417726710304189918942477, 3.88683304312991433176858606112, 4.44642014655177608914834542683, 5.53015103498019650402993804875, 5.78732227227171010271350929200, 6.87063993054943394652772627782, 7.57884293012718184499153679027, 8.45037356865529025004238788295, 9.55845749859105170586145286798, 10.508956116925095478487629415361, 11.02163274850482372474307982885, 11.36915637707140857316139868101, 12.20479694927843117331333210472, 13.072279139241911065103157831121, 14.22513255934227536692458365410, 14.66096395160104813110753849027, 14.87141555809546238781839359856, 15.78254084129263976655828083795, 16.56067098257232481217689417100, 17.18928577680974961669925809395, 17.91625208573045188999812479847, 18.88140860833991509747336291758, 19.8317876808738562312347127773

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