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Properties

Label 1-277-277.144-r0-0-0
Degree $1$
Conductor $277$
Sign $0.431 + 0.901i$
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  + (−0.917 + 0.398i)2-s + (−0.648 − 0.761i)3-s + (0.682 − 0.730i)4-s + (−0.998 + 0.0455i)5-s + (0.898 + 0.439i)6-s + (0.538 − 0.842i)7-s + (−0.334 + 0.942i)8-s + (−0.158 + 0.987i)9-s + (0.898 − 0.439i)10-s + (−0.949 + 0.313i)11-s + (−0.998 − 0.0455i)12-s + (−0.775 + 0.631i)13-s + (−0.158 + 0.987i)14-s + (0.682 + 0.730i)15-s + (−0.0682 − 0.997i)16-s + (0.0227 − 0.999i)17-s + ⋯
L(s)  = 1  + (−0.917 + 0.398i)2-s + (−0.648 − 0.761i)3-s + (0.682 − 0.730i)4-s + (−0.998 + 0.0455i)5-s + (0.898 + 0.439i)6-s + (0.538 − 0.842i)7-s + (−0.334 + 0.942i)8-s + (−0.158 + 0.987i)9-s + (0.898 − 0.439i)10-s + (−0.949 + 0.313i)11-s + (−0.998 − 0.0455i)12-s + (−0.775 + 0.631i)13-s + (−0.158 + 0.987i)14-s + (0.682 + 0.730i)15-s + (−0.0682 − 0.997i)16-s + (0.0227 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.431 + 0.901i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.431 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2700430685 + 0.1700820343i\)
\(L(\frac12)\) \(\approx\) \(0.2700430685 + 0.1700820343i\)
\(L(1)\) \(\approx\) \(0.4359222073 + 0.003653455051i\)
\(L(1)\) \(\approx\) \(0.4359222073 + 0.003653455051i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.917 + 0.398i)T \)
3 \( 1 + (-0.648 - 0.761i)T \)
5 \( 1 + (-0.998 + 0.0455i)T \)
7 \( 1 + (0.538 - 0.842i)T \)
11 \( 1 + (-0.949 + 0.313i)T \)
13 \( 1 + (-0.775 + 0.631i)T \)
17 \( 1 + (0.0227 - 0.999i)T \)
19 \( 1 + (0.460 + 0.887i)T \)
23 \( 1 + (-0.998 + 0.0455i)T \)
29 \( 1 + (-0.974 + 0.225i)T \)
31 \( 1 + (0.538 + 0.842i)T \)
37 \( 1 + (0.682 - 0.730i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (0.803 + 0.595i)T \)
47 \( 1 + (-0.877 + 0.480i)T \)
53 \( 1 + (0.995 + 0.0909i)T \)
59 \( 1 + (0.203 + 0.979i)T \)
61 \( 1 + (0.203 + 0.979i)T \)
67 \( 1 + (-0.829 + 0.557i)T \)
71 \( 1 + (0.983 - 0.181i)T \)
73 \( 1 + (-0.0682 + 0.997i)T \)
79 \( 1 + (0.538 + 0.842i)T \)
83 \( 1 + (0.898 + 0.439i)T \)
89 \( 1 + (-0.247 + 0.968i)T \)
97 \( 1 + (-0.648 - 0.761i)T \)
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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

−26.03044809793733534194262411688, −24.41651484458707656584975177829, −23.91548813066575033442105642344, −22.45278781999022404410823469299, −21.80119036902082519952494077089, −20.86335835382321062008606767281, −20.06484651930259773716360634269, −19.01958900211480775078900514603, −18.12161085062749545705835995892, −17.331154682192659237338378392803, −16.29524102147667002817263681218, −15.44953464214461766521651998735, −15.023171477690962501368968417622, −12.82165447837139323803954909805, −11.97114842859068703210140001973, −11.26082426940113444869602714202, −10.47027514934455593343777781254, −9.41105029296067675864410687931, −8.304892676031814142849136227804, −7.60930745610239562847052224848, −6.03449988881993530617258420488, −4.868918908702695658083921153, −3.61639099867143782484569348118, −2.427200298051864183178266549273, −0.37855545900654277113351261526, 1.06079452775019386826609437918, 2.45766985062132655825140677222, 4.481743959324203349167528964620, 5.54351253856017746818635435533, 7.01531955908950625286523776539, 7.529412517337828853049813919, 8.14217467926712798444119226759, 9.77388478787124693438684367010, 10.8049380982468184795238116529, 11.5509880789863909444638028070, 12.3914296834999769552691214865, 13.89251541608587737859317602880, 14.77905989802418279540847389737, 16.26942944940452913127347794767, 16.41739913665884229403977340824, 17.81757995283163379090249608927, 18.24629476077171893299506019449, 19.30371282881723439168858522931, 19.99103433518158375683098289777, 20.96591101936830274425775063757, 22.74373365847768840940378941547, 23.31911083303008884054765494030, 24.16582446813255849328452892264, 24.610996062565618323955302118831, 26.00387740250571262802754998247

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