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Properties

Label 1-277-277.68-r1-0-0
Degree $1$
Conductor $277$
Sign $-0.0791 - 0.996i$
Analytic cond. $29.7677$
Root an. cond. $29.7677$
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  + (−0.942 + 0.334i)2-s + (−0.877 + 0.480i)3-s + (0.775 − 0.631i)4-s + (−0.926 + 0.377i)5-s + (0.665 − 0.746i)6-s + (−0.613 − 0.789i)7-s + (−0.519 + 0.854i)8-s + (0.538 − 0.842i)9-s + (0.746 − 0.665i)10-s + (0.907 + 0.419i)11-s + (−0.377 + 0.926i)12-s + (−0.460 + 0.887i)13-s + (0.842 + 0.538i)14-s + (0.631 − 0.775i)15-s + (0.203 − 0.979i)16-s + (0.557 + 0.829i)17-s + ⋯
L(s)  = 1  + (−0.942 + 0.334i)2-s + (−0.877 + 0.480i)3-s + (0.775 − 0.631i)4-s + (−0.926 + 0.377i)5-s + (0.665 − 0.746i)6-s + (−0.613 − 0.789i)7-s + (−0.519 + 0.854i)8-s + (0.538 − 0.842i)9-s + (0.746 − 0.665i)10-s + (0.907 + 0.419i)11-s + (−0.377 + 0.926i)12-s + (−0.460 + 0.887i)13-s + (0.842 + 0.538i)14-s + (0.631 − 0.775i)15-s + (0.203 − 0.979i)16-s + (0.557 + 0.829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.0791 - 0.996i$
Analytic conductor: \(29.7677\)
Root analytic conductor: \(29.7677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (1:\ ),\ -0.0791 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009163110984 + 0.009919860263i\)
\(L(\frac12)\) \(\approx\) \(0.009163110984 + 0.009919860263i\)
\(L(1)\) \(\approx\) \(0.3586993351 + 0.1444440848i\)
\(L(1)\) \(\approx\) \(0.3586993351 + 0.1444440848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.942 + 0.334i)T \)
3 \( 1 + (-0.877 + 0.480i)T \)
5 \( 1 + (-0.926 + 0.377i)T \)
7 \( 1 + (-0.613 - 0.789i)T \)
11 \( 1 + (0.907 + 0.419i)T \)
13 \( 1 + (-0.460 + 0.887i)T \)
17 \( 1 + (0.557 + 0.829i)T \)
19 \( 1 + (-0.990 + 0.136i)T \)
23 \( 1 + (0.377 + 0.926i)T \)
29 \( 1 + (-0.934 + 0.356i)T \)
31 \( 1 + (-0.789 - 0.613i)T \)
37 \( 1 + (0.631 + 0.775i)T \)
41 \( 1 + (0.0682 + 0.997i)T \)
43 \( 1 + (0.761 + 0.648i)T \)
47 \( 1 + (-0.898 - 0.439i)T \)
53 \( 1 + (-0.699 + 0.715i)T \)
59 \( 1 + (0.576 - 0.816i)T \)
61 \( 1 + (0.816 + 0.576i)T \)
67 \( 1 + (-0.949 + 0.313i)T \)
71 \( 1 + (0.0227 - 0.999i)T \)
73 \( 1 + (0.979 - 0.203i)T \)
79 \( 1 + (0.613 - 0.789i)T \)
83 \( 1 + (-0.746 - 0.665i)T \)
89 \( 1 + (0.974 - 0.225i)T \)
97 \( 1 + (-0.480 - 0.877i)T \)
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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

−25.51693092437236055734709342825, −24.81915224242151961564568449013, −24.10565432722162791166424754209, −22.758236200430727582267414082162, −22.221532051061375518781861766327, −21.00859325591618361994960266151, −19.80715322780749328440651592453, −19.14788733882605361520935738928, −18.52963868321302708986094289580, −17.41099948271807812978072515476, −16.54999218158947703036731937479, −15.98050104827366839314535766071, −14.86330600778798793785441977144, −12.80025362101059365005036102612, −12.43088482961552861479177506458, −11.53494845566760528877010605091, −10.74614113572398624706186158755, −9.44839466725772128034297900850, −8.498897976013758949642012853465, −7.45519308934103241139412506296, −6.54998019130646045716053190897, −5.37284781437475904950515280439, −3.77070789367340658110761936228, −2.42053836371982727766469401443, −0.81946400454501192243354250352, 0.00954125343940219711662653883, 1.43709957396096146268117636227, 3.53794200931839511719657446698, 4.49089508839131353134531790752, 6.14468154305089847545783534063, 6.853920803989035283555988128862, 7.69470883150165678414662099783, 9.218872170739172013211878135195, 9.96457708630321243888862267319, 10.980667447277523725311399823819, 11.60668073529698439192562868253, 12.68753536434849477599428625807, 14.64330599552484696625395059345, 15.09388396942588899142295260696, 16.44271867862294846898520331933, 16.70009301053247857435282515410, 17.60108607445123869513680981401, 18.89823906928087635681427692192, 19.45053526419550465908520415381, 20.3822281482942464916669638552, 21.64000759331947278248872078500, 22.7004375443286489768264525791, 23.56776119307054037136867325944, 23.97548884156430737473128034277, 25.540551752963112700615817299169

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