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Properties

Label 1-277-277.121-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.0825 + 0.996i$
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  + (−0.962 + 0.269i)2-s + (−0.974 − 0.225i)3-s + (0.854 − 0.519i)4-s + (0.949 + 0.313i)5-s + (0.998 − 0.0455i)6-s + (0.746 + 0.665i)7-s + (−0.682 + 0.730i)8-s + (0.898 + 0.439i)9-s + (−0.998 − 0.0455i)10-s + (−0.613 + 0.789i)11-s + (−0.949 + 0.313i)12-s + (−0.0682 + 0.997i)13-s + (−0.898 − 0.439i)14-s + (−0.854 − 0.519i)15-s + (0.460 − 0.887i)16-s + (0.158 + 0.987i)17-s + ⋯
L(s)  = 1  + (−0.962 + 0.269i)2-s + (−0.974 − 0.225i)3-s + (0.854 − 0.519i)4-s + (0.949 + 0.313i)5-s + (0.998 − 0.0455i)6-s + (0.746 + 0.665i)7-s + (−0.682 + 0.730i)8-s + (0.898 + 0.439i)9-s + (−0.998 − 0.0455i)10-s + (−0.613 + 0.789i)11-s + (−0.949 + 0.313i)12-s + (−0.0682 + 0.997i)13-s + (−0.898 − 0.439i)14-s + (−0.854 − 0.519i)15-s + (0.460 − 0.887i)16-s + (0.158 + 0.987i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0825 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0825 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.0825 + 0.996i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.0825 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4377124360 + 0.4754675384i\)
\(L(\frac12)\) \(\approx\) \(0.4377124360 + 0.4754675384i\)
\(L(1)\) \(\approx\) \(0.5965768284 + 0.2202529835i\)
\(L(1)\) \(\approx\) \(0.5965768284 + 0.2202529835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.962 + 0.269i)T \)
3 \( 1 + (-0.974 - 0.225i)T \)
5 \( 1 + (0.949 + 0.313i)T \)
7 \( 1 + (0.746 + 0.665i)T \)
11 \( 1 + (-0.613 + 0.789i)T \)
13 \( 1 + (-0.0682 + 0.997i)T \)
17 \( 1 + (0.158 + 0.987i)T \)
19 \( 1 + (0.203 - 0.979i)T \)
23 \( 1 + (-0.949 - 0.313i)T \)
29 \( 1 + (0.0227 - 0.999i)T \)
31 \( 1 + (-0.746 + 0.665i)T \)
37 \( 1 + (-0.854 + 0.519i)T \)
41 \( 1 + (-0.775 + 0.631i)T \)
43 \( 1 + (0.247 - 0.968i)T \)
47 \( 1 + (0.934 + 0.356i)T \)
53 \( 1 + (-0.803 + 0.595i)T \)
59 \( 1 + (-0.990 + 0.136i)T \)
61 \( 1 + (0.990 - 0.136i)T \)
67 \( 1 + (0.538 + 0.842i)T \)
71 \( 1 + (0.291 + 0.956i)T \)
73 \( 1 + (-0.460 - 0.887i)T \)
79 \( 1 + (0.746 - 0.665i)T \)
83 \( 1 + (-0.998 + 0.0455i)T \)
89 \( 1 + (0.983 - 0.181i)T \)
97 \( 1 + (0.974 + 0.225i)T \)
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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

−25.50982176545837594041231201796, −24.525912724802731805323820987407, −23.8598195667795805489965868750, −22.51970907933336375254668276738, −21.56919955413781751973599943744, −20.79572764254413186167272546384, −20.217890120762622703404601001955, −18.50773450851664087367775230236, −18.06254964441902188216009568399, −17.2450880585417930640852799589, −16.50725608157930180336335783036, −15.75570514403372252579171861288, −14.21605185045418628354172990599, −13.02008774418984908782694365788, −12.03989457015586506429843023892, −10.92230500976855999179952951420, −10.37578676726998983401286381182, −9.522378938002562792420391192755, −8.19750051323928101205205571743, −7.24622993991261236761132863211, −5.9051734239543947177591761042, −5.16642192883778562806590263677, −3.48973096229447920846575481959, −1.838284720287660972875280553553, −0.68395101305062933252446041664, 1.6330414503847328812295284880, 2.25201429126334792636064361559, 4.793509534441204583403606888331, 5.73148421737672112561292676380, 6.585155354103064193831560584416, 7.538144538739679419025097094623, 8.804388364180523331695130377923, 9.89750988967883998715966270270, 10.64613693455411532978602230013, 11.5926274828675073266150241644, 12.50680478824131559443493055552, 13.9241096566717316464873433173, 15.05388503691215062612555211540, 15.90799678023320984238960073671, 17.18128835314343503683457321329, 17.53187211309797886634214619057, 18.39752886955893848820323778464, 18.97125561638575052593862272234, 20.479744929314783489972803414773, 21.45794701081848109111395551500, 22.06421936613134334983822232542, 23.58289654487724184938682386083, 24.10826491427156176955132129655, 25.05564898596586094593722499226, 25.9337761296015706333299664752

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