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Properties

Label 1-247-247.139-r0-0-0
Degree $1$
Conductor $247$
Sign $0.806 - 0.591i$
Analytic cond. $1.14706$
Root an. cond. $1.14706$
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  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)6-s + 7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.939 + 0.342i)5-s + (0.173 + 0.984i)6-s + 7-s + (−0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)10-s + 11-s + (−0.5 − 0.866i)12-s + (−0.939 + 0.342i)14-s + (0.173 + 0.984i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 247 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.806 - 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(247\)    =    \(13 \cdot 19\)
Sign: $0.806 - 0.591i$
Analytic conductor: \(1.14706\)
Root analytic conductor: \(1.14706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{247} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 247,\ (0:\ ),\ 0.806 - 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7774457721 - 0.2543755773i\)
\(L(\frac12)\) \(\approx\) \(0.7774457721 - 0.2543755773i\)
\(L(1)\) \(\approx\) \(0.7499221228 - 0.1241621855i\)
\(L(1)\) \(\approx\) \(0.7499221228 - 0.1241621855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.939 + 0.342i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.939 + 0.342i)T \)
7 \( 1 + T \)
11 \( 1 + T \)
17 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.766 - 0.642i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (0.173 + 0.984i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (0.766 + 0.642i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−26.71720381618123901782920588751, −25.34802089572365443684892324790, −24.66726563929281819095433681341, −23.42715484239586796984616923946, −22.23628485498055648671786311381, −21.26530427099323782762130343250, −20.45518038471750605016776322562, −19.89080113176713419361567489081, −18.89118537348604834616567954291, −17.71818914597225370474623154309, −16.69419346275724571931740777785, −16.16400150870240268912705496175, −15.038331691412167557152868991771, −14.2800373435368542498433976109, −12.33990308576554420351505184429, −11.56581942432034623151260962526, −10.86239196473363175302308806128, −9.72616136807493528200690472431, −8.667688527946255663226069774883, −8.152027371251906936615389599314, −6.877469664716242332757511882176, −5.04594831486373241649719933898, −4.02152176300719463618179716125, −2.944742499244788196307651672705, −1.19680527518388936405879555921, 0.996500163719847230419085792606, 2.13527591194152750084759607000, 3.71999143508138399227487763733, 5.55433179193486540759433281451, 6.72694456301217575790422483683, 7.62320905453718798955871918307, 8.20131502758316332329623387189, 9.22759894244285590892100068979, 10.80681793362592883249344920713, 11.59879392933183094287987453622, 12.283663231439876068240806337520, 14.09790855029098529409927507052, 14.67718047560722090413898107275, 15.6393043494252052426848375501, 17.04890114996784018544555331383, 17.571685868027250220935816634922, 18.68400837829856956753610719178, 19.2768197467185546386233379942, 19.99962412091529222182706262798, 21.03586922854558186468186539594, 22.66930523669937644294099377371, 23.732773061168385526181329631426, 24.12586292742325466076271329317, 25.13036556775650582980571705333, 25.89608842393242078993162102043

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