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Properties

Label 1-277-277.185-r0-0-0
Degree $1$
Conductor $277$
Sign $0.343 - 0.939i$
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  + (0.682 − 0.730i)2-s + (0.934 − 0.356i)3-s + (−0.0682 − 0.997i)4-s + (−0.419 + 0.907i)5-s + (0.377 − 0.926i)6-s + (0.995 + 0.0909i)7-s + (−0.775 − 0.631i)8-s + (0.746 − 0.665i)9-s + (0.377 + 0.926i)10-s + (0.113 + 0.993i)11-s + (−0.419 − 0.907i)12-s + (0.203 − 0.979i)13-s + (0.746 − 0.665i)14-s + (−0.0682 + 0.997i)15-s + (−0.990 + 0.136i)16-s + (0.538 − 0.842i)17-s + ⋯
L(s)  = 1  + (0.682 − 0.730i)2-s + (0.934 − 0.356i)3-s + (−0.0682 − 0.997i)4-s + (−0.419 + 0.907i)5-s + (0.377 − 0.926i)6-s + (0.995 + 0.0909i)7-s + (−0.775 − 0.631i)8-s + (0.746 − 0.665i)9-s + (0.377 + 0.926i)10-s + (0.113 + 0.993i)11-s + (−0.419 − 0.907i)12-s + (0.203 − 0.979i)13-s + (0.746 − 0.665i)14-s + (−0.0682 + 0.997i)15-s + (−0.990 + 0.136i)16-s + (0.538 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.343 - 0.939i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.343 - 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.947450879 - 1.361373050i\)
\(L(\frac12)\) \(\approx\) \(1.947450879 - 1.361373050i\)
\(L(1)\) \(\approx\) \(1.729762750 - 0.8337701184i\)
\(L(1)\) \(\approx\) \(1.729762750 - 0.8337701184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (0.682 - 0.730i)T \)
3 \( 1 + (0.934 - 0.356i)T \)
5 \( 1 + (-0.419 + 0.907i)T \)
7 \( 1 + (0.995 + 0.0909i)T \)
11 \( 1 + (0.113 + 0.993i)T \)
13 \( 1 + (0.203 - 0.979i)T \)
17 \( 1 + (0.538 - 0.842i)T \)
19 \( 1 + (-0.576 + 0.816i)T \)
23 \( 1 + (-0.419 + 0.907i)T \)
29 \( 1 + (-0.829 - 0.557i)T \)
31 \( 1 + (0.995 - 0.0909i)T \)
37 \( 1 + (-0.0682 - 0.997i)T \)
41 \( 1 + (0.460 + 0.887i)T \)
43 \( 1 + (-0.974 - 0.225i)T \)
47 \( 1 + (-0.998 - 0.0455i)T \)
53 \( 1 + (-0.648 + 0.761i)T \)
59 \( 1 + (-0.917 + 0.398i)T \)
61 \( 1 + (-0.917 + 0.398i)T \)
67 \( 1 + (0.613 + 0.789i)T \)
71 \( 1 + (-0.158 + 0.987i)T \)
73 \( 1 + (-0.990 - 0.136i)T \)
79 \( 1 + (0.995 - 0.0909i)T \)
83 \( 1 + (0.377 - 0.926i)T \)
89 \( 1 + (0.0227 + 0.999i)T \)
97 \( 1 + (0.934 - 0.356i)T \)
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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

−25.833053658127174494388409977279, −24.581469925265954705411980689293, −24.21075387379481196845016413165, −23.55497045353628218024947043144, −21.98243681534090584248186663849, −21.23311570952944481094425856831, −20.7526763276545858885130059226, −19.63661717342526197895016313780, −18.587826189610676362128814895671, −17.07418297972488018246745773780, −16.49419744754059512874644904997, −15.54175084630026167510578657901, −14.66674539318223224691825797829, −13.909093034135361285851835551844, −13.082350264738669158136333173268, −11.96205037813086663147203922745, −10.92032884031496693923522117001, −9.14319123907083448375320889815, −8.396559112760001394737183844682, −7.914097844838759318999987043801, −6.47889951778917472774945073839, −5.00973111428612701079190145919, −4.34183021616684908261719456092, −3.3914466315543770673254646208, −1.78797573313503660190110190138, 1.52911591385429807192229748763, 2.53254953894508572323029840432, 3.53602592496496314577795885117, 4.532271108854850960477028584878, 5.960197906438529851530921236272, 7.32680804782528194815753798366, 8.05888554628985360477004186659, 9.582968759232493563444351210226, 10.39355892056015836952746551138, 11.558442009744985391355241226495, 12.3179144110131887199138621955, 13.443923339240053822159913708, 14.379377049470232154627580641836, 14.92751318360899821137167192377, 15.59199820507929250364417484437, 17.77417971108539806459886370771, 18.370420023304677281334249963911, 19.252292739858171755243813584178, 20.178661645006247110013017160288, 20.79577206688928684595076077713, 21.68562174600045415275384998364, 22.989234108334568638437751491, 23.31182462073549504417916921501, 24.65277134122841139674356122909, 25.21750339277442479481103159184

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