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Properties

Label 1-277-277.169-r0-0-0
Degree $1$
Conductor $277$
Sign $0.0421 - 0.999i$
Analytic cond. $1.28638$
Root an. cond. $1.28638$
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  + (−0.990 + 0.136i)2-s + (−0.917 + 0.398i)3-s + (0.962 − 0.269i)4-s + (−0.775 − 0.631i)5-s + (0.854 − 0.519i)6-s + (−0.775 + 0.631i)7-s + (−0.917 + 0.398i)8-s + (0.682 − 0.730i)9-s + (0.854 + 0.519i)10-s + (−0.0682 + 0.997i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.682 − 0.730i)14-s + (0.962 + 0.269i)15-s + (0.854 − 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯
L(s)  = 1  + (−0.990 + 0.136i)2-s + (−0.917 + 0.398i)3-s + (0.962 − 0.269i)4-s + (−0.775 − 0.631i)5-s + (0.854 − 0.519i)6-s + (−0.775 + 0.631i)7-s + (−0.917 + 0.398i)8-s + (0.682 − 0.730i)9-s + (0.854 + 0.519i)10-s + (−0.0682 + 0.997i)11-s + (−0.775 + 0.631i)12-s + (0.682 + 0.730i)13-s + (0.682 − 0.730i)14-s + (0.962 + 0.269i)15-s + (0.854 − 0.519i)16-s + (−0.334 − 0.942i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0421 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0421 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.0421 - 0.999i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.0421 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1364722691 - 0.1308309266i\)
\(L(\frac12)\) \(\approx\) \(0.1364722691 - 0.1308309266i\)
\(L(1)\) \(\approx\) \(0.3666788859 + 0.02833542323i\)
\(L(1)\) \(\approx\) \(0.3666788859 + 0.02833542323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.990 + 0.136i)T \)
3 \( 1 + (-0.917 + 0.398i)T \)
5 \( 1 + (-0.775 - 0.631i)T \)
7 \( 1 + (-0.775 + 0.631i)T \)
11 \( 1 + (-0.0682 + 0.997i)T \)
13 \( 1 + (0.682 + 0.730i)T \)
17 \( 1 + (-0.334 - 0.942i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (-0.775 - 0.631i)T \)
29 \( 1 + (0.962 + 0.269i)T \)
31 \( 1 + (-0.775 - 0.631i)T \)
37 \( 1 + (0.962 - 0.269i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (-0.990 + 0.136i)T \)
47 \( 1 + (-0.334 - 0.942i)T \)
53 \( 1 + (0.203 - 0.979i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (-0.0682 - 0.997i)T \)
67 \( 1 + (0.854 - 0.519i)T \)
71 \( 1 + (-0.917 + 0.398i)T \)
73 \( 1 + (0.854 + 0.519i)T \)
79 \( 1 + (-0.775 - 0.631i)T \)
83 \( 1 + (0.854 - 0.519i)T \)
89 \( 1 + (-0.576 - 0.816i)T \)
97 \( 1 + (-0.917 + 0.398i)T \)
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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

−26.06630967058202812496568303593, −25.16262202161633819159672578745, −23.73559340048629132701460593312, −23.54134329886989076859865190271, −22.21071133565875149925694831738, −21.48129075440013492173412317840, −19.80226772205679539908791235598, −19.49440256406925610823857902894, −18.4926804700449095874346905150, −17.76813700691119671228455089998, −16.71849396865398968992224288771, −16.03444646211643707568504604305, −15.235086285420977257667832285155, −13.482694556728007953779470479443, −12.556059760104587716799136388234, −11.414508129624552568430172136620, −10.784527462012849825887353932458, −10.14831345495221124478333248495, −8.491162559430548192652175247866, −7.66821160487405843426060687740, −6.57686019978434039277612917118, −6.0508608739295934533300987096, −3.98616228480257009041088307303, −2.88480355882286092289983168088, −1.08219834501878741169744430879, 0.234553170986103056668931185968, 1.95702249096163180456709738342, 3.76751634194928998648897514696, 4.99665360070601502522476399602, 6.25266489612913280160196700421, 7.02168411277050132636508467044, 8.392494624913789818583004657196, 9.325511477826767828832634088213, 10.112315328067564320753668117917, 11.32273088795648346796470795729, 12.03855914038071610765906254022, 12.7763400813810187467535432366, 14.832135199194614282554453552373, 15.784902874694180319757896131406, 16.210879587924621487889186884392, 16.9889404397869588232985254319, 18.203809383576290181143690804837, 18.747410942531526816313387749068, 19.96326069435680033559138241960, 20.67809523140314020996229414611, 21.74205027775043076216747043833, 22.99133697115127465572826637360, 23.55147792297060110471059074005, 24.63426190843498071072474412262, 25.54577181047071594984737134506

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