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Properties

Label 1-277-277.190-r0-0-0
Degree $1$
Conductor $277$
Sign $0.342 + 0.939i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.282179914 + 0.8974554763i\)
\(L(\frac12)\) \(\approx\) \(1.282179914 + 0.8974554763i\)
\(L(1)\) \(\approx\) \(1.271176605 + 0.4707030105i\)
\(L(1)\) \(\approx\) \(1.271176605 + 0.4707030105i\)

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

−24.68062588925539412304394304877, −24.44040399637302633925300841290, −23.67072923678452765632413800703, −22.78303330508513185106371697360, −21.91754511992634136857651773297, −21.2894601125537903230596527613, −20.10016778678787754712982887509, −19.12192381890004584051529619871, −18.36845898138415541937630948876, −16.93473465636655466941237038667, −16.09976235051715404038600832982, −15.39553060416873993406147088434, −14.2107341388892587381559061434, −13.23725049510341392272364330077, −11.8709059956709733032876568473, −11.732982611054462077358154740401, −11.053378391519661349226689682653, −9.4334349523139500916357705552, −7.97141813112546141260230117165, −6.84603041349185419699992223154, −5.883854662523796609950730297768, −4.74978710055647937573038086505, −4.16849381813400552439532456492, −2.46947507562608067286696380147, −1.00457855156891935040072735521, 1.54386942806241507061637198021, 3.57814772722566495675207434777, 4.24637751103907008965913285107, 5.21065440686934772174165304464, 6.3982104605417584377704645166, 7.34576338344766206361441152950, 8.13952011292732681506373611591, 10.33900106305950671634947255585, 10.843683316240690562706709988418, 12.11277741178417138897054116775, 12.2953954398127763761834219442, 13.8256552191683178842307269465, 14.96697465366687666540423929508, 15.40951301704427865348831007424, 16.58920346760501577115241533035, 17.28470276560355918159205232763, 18.25721910476610955903261888515, 19.88197075708885801674419028058, 20.40386735790072205797538452384, 21.64116253083206995865776366724, 22.42976028153868066863192868656, 23.16865993946951180869603538570, 23.70353520685356829242085899745, 24.53534959254064129689161611405, 25.71013471276686753978961054893

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