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Properties

Label 1-277-277.193-r0-0-0
Degree $1$
Conductor $277$
Sign $0.0218 + 0.999i$
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 + (−0.334 + 0.942i)3-s + (0.682 − 0.730i)4-s + (−0.460 + 0.887i)5-s + (0.0682 + 0.997i)6-s + (0.460 + 0.887i)7-s + (0.334 − 0.942i)8-s + (−0.775 − 0.631i)9-s + (−0.0682 + 0.997i)10-s + (−0.203 + 0.979i)11-s + (0.460 + 0.887i)12-s + (−0.775 + 0.631i)13-s + (0.775 + 0.631i)14-s + (−0.682 − 0.730i)15-s + (−0.0682 − 0.997i)16-s + (−0.854 − 0.519i)17-s + ⋯
L(s)  = 1  + (0.917 − 0.398i)2-s + (−0.334 + 0.942i)3-s + (0.682 − 0.730i)4-s + (−0.460 + 0.887i)5-s + (0.0682 + 0.997i)6-s + (0.460 + 0.887i)7-s + (0.334 − 0.942i)8-s + (−0.775 − 0.631i)9-s + (−0.0682 + 0.997i)10-s + (−0.203 + 0.979i)11-s + (0.460 + 0.887i)12-s + (−0.775 + 0.631i)13-s + (0.775 + 0.631i)14-s + (−0.682 − 0.730i)15-s + (−0.0682 − 0.997i)16-s + (−0.854 − 0.519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0218 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0218 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.0218 + 0.999i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.0218 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.144546291 + 1.119758042i\)
\(L(\frac12)\) \(\approx\) \(1.144546291 + 1.119758042i\)
\(L(1)\) \(\approx\) \(1.328273788 + 0.4948554621i\)
\(L(1)\) \(\approx\) \(1.328273788 + 0.4948554621i\)

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 \( 1 + (-0.334 + 0.942i)T \)
5 \( 1 + (-0.460 + 0.887i)T \)
7 \( 1 + (0.460 + 0.887i)T \)
11 \( 1 + (-0.203 + 0.979i)T \)
13 \( 1 + (-0.775 + 0.631i)T \)
17 \( 1 + (-0.854 - 0.519i)T \)
19 \( 1 + (0.460 + 0.887i)T \)
23 \( 1 + (0.460 - 0.887i)T \)
29 \( 1 + (0.682 + 0.730i)T \)
31 \( 1 + (-0.460 + 0.887i)T \)
37 \( 1 + (-0.682 + 0.730i)T \)
41 \( 1 + (0.854 + 0.519i)T \)
43 \( 1 + (0.917 - 0.398i)T \)
47 \( 1 + (0.854 + 0.519i)T \)
53 \( 1 + (0.576 - 0.816i)T \)
59 \( 1 + (0.203 + 0.979i)T \)
61 \( 1 + (-0.203 - 0.979i)T \)
67 \( 1 + (-0.0682 - 0.997i)T \)
71 \( 1 + (-0.334 + 0.942i)T \)
73 \( 1 + (0.0682 - 0.997i)T \)
79 \( 1 + (0.460 - 0.887i)T \)
83 \( 1 + (-0.0682 - 0.997i)T \)
89 \( 1 + (0.962 - 0.269i)T \)
97 \( 1 + (0.334 - 0.942i)T \)
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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

−24.85595221602201464956301890451, −24.29861817538142956049604896518, −23.83685915200630351961442682609, −23.01407573610444451210494921465, −22.02303743863714934031153925859, −20.94067811067941144964833989772, −19.9140907720531715351516046874, −19.41866247879323960896788929076, −17.578449435762972817653746060842, −17.22777548766542065129368996607, −16.214448201740337206127320032129, −15.24368669730810228948446054689, −13.90046506295400337709504711661, −13.33531277912907702985128870337, −12.556810308912750664222895385942, −11.51396636812634765950878924360, −10.87769984379906197154537450715, −8.75734709875375700864028245534, −7.756810452865128881952935966425, −7.219686655998725128534920378588, −5.82995536270112165925945750525, −5.01982146549827337883563850672, −3.91102609823879861461744260239, −2.43943470971712916613412031326, −0.83474264563923550834726354584, 2.19082167343216895469650722889, 3.11032559172659115302706778063, 4.42608790674073967397173254861, 5.030754756254534384089561184980, 6.299919885376724711650195229043, 7.300229726492186678571592876367, 9.02750215522092647184518414685, 10.13714079811436150227562768723, 10.90494530021893297486818117183, 11.876027813067103702695903387051, 12.366757456969444945625024020987, 14.23431225215201099005461892731, 14.68455284112793351773788321613, 15.492759857821371587902229526979, 16.20914696913716442166764075532, 17.73384166176671421942808509679, 18.65212076813707165493800087491, 19.7858801762029411488480655494, 20.67676312625790740256666687838, 21.53242300302751573939275873104, 22.33876780393217912925863972387, 22.765739665492158746500876440331, 23.77852864531580307350143181696, 24.84613896943366003284655582419, 25.872978600154832341163687537166

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