Properties

Label 1-7e2-49.4-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

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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: $\chi_{49} (4, \cdot )$
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 \( 1 + T \)
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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

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

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