↑

Properties

Label 1-7e2-49.37-r0-0-0
Degree $1$
Conductor $49$
Sign $-0.304 - 0.952i$
Analytic cond. $0.227555$
Root an. cond. $0.227555$
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  + (0.365 − 0.930i)2-s + (0.0747 − 0.997i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−0.900 − 0.433i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (0.826 − 0.563i)10-s + (−0.988 + 0.149i)11-s + (−0.733 + 0.680i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (0.0747 − 0.997i)3-s + (−0.733 − 0.680i)4-s + (0.826 + 0.563i)5-s + (−0.900 − 0.433i)6-s + (−0.900 + 0.433i)8-s + (−0.988 − 0.149i)9-s + (0.826 − 0.563i)10-s + (−0.988 + 0.149i)11-s + (−0.733 + 0.680i)12-s + (0.623 + 0.781i)13-s + (0.623 − 0.781i)15-s + (0.0747 + 0.997i)16-s + (0.955 + 0.294i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ -0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5984243830 - 0.8199696252i\)
\(L(\frac12)\) \(\approx\) \(0.5984243830 - 0.8199696252i\)
\(L(1)\) \(\approx\) \(0.8916385770 - 0.7449344732i\)
\(L(1)\) \(\approx\) \(0.8916385770 - 0.7449344732i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (0.365 - 0.930i)T \)
3 \( 1 + (0.0747 - 0.997i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.955 + 0.294i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.955 - 0.294i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.733 + 0.680i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (0.365 - 0.930i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (0.826 - 0.563i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 - 0.781i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + T \)
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

−33.769060600612141242996529687606, −33.108131413381993551299321237751, −32.10009239280642100185618981517, −31.35753258368003250664475993219, −29.59496418732931667753582913475, −28.112863541047876889838678595860, −27.12623361146201607273173551152, −25.755474286790272925276889232565, −25.25014505331386761085533042029, −23.64563006189079009581181594353, −22.556732979567965955793729019377, −21.25270333341109862936935681645, −20.69913332560368789735333618930, −18.36584733802690021731313434337, −17.04602673250848187318496689297, −16.21095603020236584334506987214, −15.0582957357044940337160828207, −13.82519607851658695958265172121, −12.66000538825105689098883078416, −10.49266139739859118748639549972, −9.19823688210446215948040461309, −8.01496299957539216945930909694, −5.81882298431529564614007919433, −5.05453872317855741989125462184, −3.30967935053175643034747760439, 1.75701721334817229031702603678, 3.03204375553334677159122581725, 5.36405405178046692028597137915, 6.73365680103489211070153485326, 8.68983917992256112759209871115, 10.26534750920651201489718129464, 11.48458996386862395670432042539, 12.95098472342750903408666705387, 13.68084495349752417434520182064, 14.8608718513470266684956922002, 17.28531972971915420707774003981, 18.46331783756790540199991846386, 19.055478674330229419360850843728, 20.64675405652888272044759049189, 21.6073322607988940190262877773, 23.046776226755280420480523620232, 23.82133485377598434138232609768, 25.39316866813412486868160001837, 26.454335320779758980562610892826, 28.37476902238190696801206492667, 29.033577081553473706021538981161, 30.1402219983582858521660891117, 30.80379474396644190108736685772, 32.01806312998474638439607965271, 33.40752300900522123664791289080

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