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Properties

Label 1-277-277.273-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.976 - 0.213i$
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  + (0.854 + 0.519i)2-s + (−0.0682 + 0.997i)3-s + (0.460 + 0.887i)4-s + (−0.917 − 0.398i)5-s + (−0.576 + 0.816i)6-s + (−0.917 + 0.398i)7-s + (−0.0682 + 0.997i)8-s + (−0.990 − 0.136i)9-s + (−0.576 − 0.816i)10-s + (0.962 − 0.269i)11-s + (−0.917 + 0.398i)12-s + (−0.990 + 0.136i)13-s + (−0.990 − 0.136i)14-s + (0.460 − 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.203 + 0.979i)17-s + ⋯
L(s)  = 1  + (0.854 + 0.519i)2-s + (−0.0682 + 0.997i)3-s + (0.460 + 0.887i)4-s + (−0.917 − 0.398i)5-s + (−0.576 + 0.816i)6-s + (−0.917 + 0.398i)7-s + (−0.0682 + 0.997i)8-s + (−0.990 − 0.136i)9-s + (−0.576 − 0.816i)10-s + (0.962 − 0.269i)11-s + (−0.917 + 0.398i)12-s + (−0.990 + 0.136i)13-s + (−0.990 − 0.136i)14-s + (0.460 − 0.887i)15-s + (−0.576 + 0.816i)16-s + (0.203 + 0.979i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.976 - 0.213i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.976 - 0.213i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1105798970 + 1.021519833i\)
\(L(\frac12)\) \(\approx\) \(-0.1105798970 + 1.021519833i\)
\(L(1)\) \(\approx\) \(0.7438143971 + 0.8091052567i\)
\(L(1)\) \(\approx\) \(0.7438143971 + 0.8091052567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (0.854 + 0.519i)T \)
3 \( 1 + (-0.0682 + 0.997i)T \)
5 \( 1 + (-0.917 - 0.398i)T \)
7 \( 1 + (-0.917 + 0.398i)T \)
11 \( 1 + (0.962 - 0.269i)T \)
13 \( 1 + (-0.990 + 0.136i)T \)
17 \( 1 + (0.203 + 0.979i)T \)
19 \( 1 + (-0.917 + 0.398i)T \)
23 \( 1 + (-0.917 - 0.398i)T \)
29 \( 1 + (0.460 - 0.887i)T \)
31 \( 1 + (-0.917 - 0.398i)T \)
37 \( 1 + (0.460 + 0.887i)T \)
41 \( 1 + (0.203 + 0.979i)T \)
43 \( 1 + (0.854 + 0.519i)T \)
47 \( 1 + (0.203 + 0.979i)T \)
53 \( 1 + (0.682 - 0.730i)T \)
59 \( 1 + (0.962 + 0.269i)T \)
61 \( 1 + (0.962 + 0.269i)T \)
67 \( 1 + (-0.576 + 0.816i)T \)
71 \( 1 + (-0.0682 + 0.997i)T \)
73 \( 1 + (-0.576 - 0.816i)T \)
79 \( 1 + (-0.917 - 0.398i)T \)
83 \( 1 + (-0.576 + 0.816i)T \)
89 \( 1 + (-0.775 + 0.631i)T \)
97 \( 1 + (-0.0682 + 0.997i)T \)
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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

−24.97792375095241263393537271348, −23.94262799839494123414830384403, −23.3280519974181759648393621837, −22.5338254455361361861084646569, −21.98008033024669251023957754445, −20.19857514384757908515109994544, −19.70136345576848212036517212389, −19.2148171124036824037773049781, −18.134195667545936802909386069293, −16.79536664239553291996662757762, −15.76735187033285778095664695294, −14.57370603449272454415870657719, −14.00297214184467741996594413327, −12.72534938967209335278056779085, −12.23068187794108206458492762200, −11.38765562323505084880920550795, −10.30537766916341359498185324086, −9.025468760382990151249228474439, −7.15024376450414031294079248881, −7.05389001061951119627880539238, −5.73172243252279848219608402033, −4.27270506717932984616778381678, −3.25676098858623055881711946449, −2.20123853246378846916201955126, −0.50063994807537643787761431550, 2.66497671777608694380385498981, 3.91168414643715237198242385398, 4.29404197453517747005242542925, 5.70114298522971551349814484476, 6.52476190859508448172119667514, 7.997118744686045133759401883511, 8.85604224260559947304284925434, 10.0260397292732527175275944516, 11.422406288005483763154631393462, 12.134047841491325292540377076372, 12.97287760007175023669675759732, 14.56306110806907086894662956704, 14.93134722332436138813113221139, 16.01285271016970640193549133203, 16.56801193529428106567172043245, 17.281000822612213180608303867729, 19.28534982744001076678145859009, 19.83455120827905768622074030263, 20.93056107670308167317541156229, 21.95920157768871926820684979921, 22.3720070022626270502759195832, 23.31880591833088580809903981627, 24.20434369145525212870895369106, 25.21881169590708430547006631346, 26.07191992653911379493069286926

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