↑

Properties

Label 1-3751-3751.235-r0-0-0
Degree $1$
Conductor $3751$
Sign $-0.0687 - 0.997i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.654 + 0.755i)2-s + (−0.978 + 0.207i)3-s + (−0.142 − 0.989i)4-s + (0.964 + 0.263i)5-s + (0.483 − 0.875i)6-s + (0.235 + 0.971i)7-s + (0.841 + 0.540i)8-s + (0.913 − 0.406i)9-s + (−0.830 + 0.556i)10-s + (0.345 + 0.938i)12-s + (−0.969 + 0.244i)13-s + (−0.888 − 0.458i)14-s + (−0.998 − 0.0570i)15-s + (−0.959 + 0.281i)16-s + (−0.398 − 0.917i)17-s + (−0.290 + 0.956i)18-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.978 + 0.207i)3-s + (−0.142 − 0.989i)4-s + (0.964 + 0.263i)5-s + (0.483 − 0.875i)6-s + (0.235 + 0.971i)7-s + (0.841 + 0.540i)8-s + (0.913 − 0.406i)9-s + (−0.830 + 0.556i)10-s + (0.345 + 0.938i)12-s + (−0.969 + 0.244i)13-s + (−0.888 − 0.458i)14-s + (−0.998 − 0.0570i)15-s + (−0.959 + 0.281i)16-s + (−0.398 − 0.917i)17-s + (−0.290 + 0.956i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0687 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0687 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $-0.0687 - 0.997i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ -0.0687 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04481024631 + 0.04800656044i\)
\(L(\frac12)\) \(\approx\) \(-0.04481024631 + 0.04800656044i\)
\(L(1)\) \(\approx\) \(0.4855163154 + 0.2892418823i\)
\(L(1)\) \(\approx\) \(0.4855163154 + 0.2892418823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.654 + 0.755i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (0.964 + 0.263i)T \)
7 \( 1 + (0.235 + 0.971i)T \)
13 \( 1 + (-0.969 + 0.244i)T \)
17 \( 1 + (-0.398 - 0.917i)T \)
19 \( 1 + (0.00951 + 0.999i)T \)
23 \( 1 + (0.610 - 0.791i)T \)
29 \( 1 + (-0.736 + 0.676i)T \)
37 \( 1 + (-0.532 + 0.846i)T \)
41 \( 1 + (0.820 + 0.572i)T \)
43 \( 1 + (-0.935 - 0.353i)T \)
47 \( 1 + (-0.921 + 0.389i)T \)
53 \( 1 + (-0.991 + 0.132i)T \)
59 \( 1 + (0.820 - 0.572i)T \)
61 \( 1 + (-0.921 + 0.389i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.217 - 0.976i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (0.449 + 0.893i)T \)
83 \( 1 + (-0.991 + 0.132i)T \)
89 \( 1 + (-0.870 + 0.491i)T \)
97 \( 1 + (0.774 - 0.633i)T \)
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

−17.79564450357780460282618591027, −17.40734494357712425856453355496, −17.18435114160768473177254772367, −16.49113306774533144846234615171, −15.63533299889948904960320992401, −14.56564640387290034583183482845, −13.580234477139804319268287280034, −13.03061512112536127274186061711, −12.709121968284859397437306162801, −11.64241237443958544529821302395, −11.112244616141403640391711610155, −10.45840393250813240393814096765, −9.90924054542782254694720150212, −9.303020347524572809517757915, −8.35007642577725566345580771872, −7.36422767248385729485823325827, −7.02279125976256679009724373164, −6.044971396944790008211692947099, −5.086387003456044529469092702207, −4.54137993013484949081149657604, −3.64278378352412007146032985243, −2.45308390445732891672263266122, −1.72910889259890795658161863472, −1.0250037765080823354568979478, −0.029257477120061293809150719364, 1.39873048506255325557527035363, 2.00789773760613644669767267033, 3.03914711643524902526729274811, 4.68802955580055914315453871845, 5.01501489005774754524998236557, 5.71317852905614926466142468225, 6.40879407951691397175972672466, 6.90209876188149944928800823247, 7.77453471477451437046285866504, 8.786568226363128787803095813089, 9.49899028150645024113361755960, 9.840298911157640978082331283504, 10.711133133159752440406328214594, 11.29549084989481392666615762010, 12.161484395028301152613250998877, 12.85804837217634602466807476035, 13.818535637199009488120339724447, 14.655524835536754957380657298681, 14.99109541155307644113158254940, 15.877496238344855676184623561810, 16.628507871739558511912578224826, 16.96400330739045171300899063936, 17.772656997673046025422713040021, 18.349985382345919343486580383727, 18.58671853335222050295696317874

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