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Properties

Label 1-7e2-49.9-r0-0-0
Degree $1$
Conductor $49$
Sign $-0.180 + 0.983i$
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  + (0.0747 + 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (0.955 + 0.294i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (−0.222 + 0.974i)6-s + (−0.222 − 0.974i)8-s + (0.826 + 0.563i)9-s + (−0.733 − 0.680i)10-s + (0.826 − 0.563i)11-s + (−0.988 − 0.149i)12-s + (−0.900 − 0.433i)13-s + (−0.900 + 0.433i)15-s + (0.955 − 0.294i)16-s + (0.365 + 0.930i)17-s + (−0.5 + 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.180 + 0.983i$
Analytic conductor: \(0.227555\)
Root analytic conductor: \(0.227555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 49,\ (0:\ ),\ -0.180 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6215422964 + 0.7461063056i\)
\(L(\frac12)\) \(\approx\) \(0.6215422964 + 0.7461063056i\)
\(L(1)\) \(\approx\) \(0.8941903223 + 0.6667823158i\)
\(L(1)\) \(\approx\) \(0.8941903223 + 0.6667823158i\)

Euler product

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

−33.050419127018849291811551559948, −31.66983281947063958025538686705, −31.46567552674061170748760079027, −30.131922132806117281779595984275, −29.15764850041286239214069589606, −27.59009391677608301906063798394, −27.03193801327451585101686334753, −25.3952546797193109641880564360, −24.13744133472030256425483977421, −22.953698928346765186115568942366, −21.41395226716727632966091140546, −20.285469144156621709858776157711, −19.61828974924320153022841107854, −18.60393323226349988787207209608, −16.953517507083550132858900930792, −15.0700471256074107720615258495, −13.9677101210085745767616755592, −12.56590217723395437377606096137, −11.809509776643290623425627911016, −9.77507150179969122072786599473, −8.823908334926926098982783810664, −7.43168652526564684449745036150, −4.699081667584628521138836757287, −3.456239851768577080933030946163, −1.66120189701890085511173596314, 3.20609248834328382833663615757, 4.50129761572855002434912630862, 6.6222983166209683958116040400, 7.84508320030159829778493793470, 8.92000199084025167245464411045, 10.46373422870415167549428655519, 12.560435470748406590762797966470, 14.15558590051888667775178865296, 14.86587965245352911219539018652, 15.85172560609709720951199265470, 17.26812926857122366541325120641, 18.94160418323464753567156365082, 19.6301004022053841555019424292, 21.57533839287893341219052462235, 22.503394330100626386590599281109, 23.94197961321592667493942042088, 24.94594218920471627265547408482, 26.12501733536833379423283729062, 26.921648412207451261975657978997, 27.73927594715139781661274750414, 30.135891608855586420349522805, 30.91330230915581376303564273965, 32.14269512298520859352989863229, 32.727204822126194225193426305633, 34.302293760013111500644272882216

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