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Properties

Label 1-277-277.164-r0-0-0
Degree $1$
Conductor $277$
Sign $-0.990 - 0.138i$
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  + (−0.334 − 0.942i)2-s + (0.854 + 0.519i)3-s + (−0.775 + 0.631i)4-s + (−0.990 − 0.136i)5-s + (0.203 − 0.979i)6-s + (−0.990 + 0.136i)7-s + (0.854 + 0.519i)8-s + (0.460 + 0.887i)9-s + (0.203 + 0.979i)10-s + (−0.576 − 0.816i)11-s + (−0.990 + 0.136i)12-s + (0.460 − 0.887i)13-s + (0.460 + 0.887i)14-s + (−0.775 − 0.631i)15-s + (0.203 − 0.979i)16-s + (−0.0682 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.334 − 0.942i)2-s + (0.854 + 0.519i)3-s + (−0.775 + 0.631i)4-s + (−0.990 − 0.136i)5-s + (0.203 − 0.979i)6-s + (−0.990 + 0.136i)7-s + (0.854 + 0.519i)8-s + (0.460 + 0.887i)9-s + (0.203 + 0.979i)10-s + (−0.576 − 0.816i)11-s + (−0.990 + 0.136i)12-s + (0.460 − 0.887i)13-s + (0.460 + 0.887i)14-s + (−0.775 − 0.631i)15-s + (0.203 − 0.979i)16-s + (−0.0682 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $-0.990 - 0.138i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ -0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02588887146 - 0.3721232552i\)
\(L(\frac12)\) \(\approx\) \(0.02588887146 - 0.3721232552i\)
\(L(1)\) \(\approx\) \(0.5922582635 - 0.2710377966i\)
\(L(1)\) \(\approx\) \(0.5922582635 - 0.2710377966i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.334 - 0.942i)T \)
3 \( 1 + (0.854 + 0.519i)T \)
5 \( 1 + (-0.990 - 0.136i)T \)
7 \( 1 + (-0.990 + 0.136i)T \)
11 \( 1 + (-0.576 - 0.816i)T \)
13 \( 1 + (0.460 - 0.887i)T \)
17 \( 1 + (-0.0682 - 0.997i)T \)
19 \( 1 + (-0.990 + 0.136i)T \)
23 \( 1 + (-0.990 - 0.136i)T \)
29 \( 1 + (-0.775 - 0.631i)T \)
31 \( 1 + (-0.990 - 0.136i)T \)
37 \( 1 + (-0.775 + 0.631i)T \)
41 \( 1 + (-0.0682 - 0.997i)T \)
43 \( 1 + (-0.334 - 0.942i)T \)
47 \( 1 + (-0.0682 - 0.997i)T \)
53 \( 1 + (0.962 - 0.269i)T \)
59 \( 1 + (-0.576 + 0.816i)T \)
61 \( 1 + (-0.576 + 0.816i)T \)
67 \( 1 + (0.203 - 0.979i)T \)
71 \( 1 + (0.854 + 0.519i)T \)
73 \( 1 + (0.203 + 0.979i)T \)
79 \( 1 + (-0.990 - 0.136i)T \)
83 \( 1 + (0.203 - 0.979i)T \)
89 \( 1 + (0.682 + 0.730i)T \)
97 \( 1 + (0.854 + 0.519i)T \)
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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

−26.116768680113018332803959888686, −25.47621419086437027785700368784, −24.18152352378509167832650422862, −23.585117442012634318918659992320, −22.99126887492669470876032124831, −21.67387139010112119761360735408, −20.15517647443783221304372458127, −19.512626681792152932160059718619, −18.81320297703263837501685077559, −18.03521046448238563319764434236, −16.68639413919804358756456531234, −15.81227820832779828604409475973, −15.084097539578177426940148466233, −14.27865595359282934420116982193, −13.052779231151166354969423716880, −12.51127548676164560726569093760, −10.76538894529536848170012695251, −9.61499064418671920549983752787, −8.70540915877550108847285699747, −7.807249816237015076972745854656, −6.987424479370181181198802161854, −6.21008802653175098741328396134, −4.353889558731614003960119768847, −3.55999202787668456243130337220, −1.81092979499276652744234269777, 0.24559350370073603635897217768, 2.37118916111263761326568818923, 3.38741805065686184197346890026, 3.93734309266831235965762366788, 5.37482222278739039465381467535, 7.361880355001560210967714943340, 8.34086779546900410156727061607, 8.95349741715206883306652746305, 10.15197920080452363732677069131, 10.832820454064246029229522527176, 12.04014732037245587577548547310, 13.06829916550793400273371873047, 13.718429908813296599188433130011, 15.179286429609786962867266209310, 15.99263655630402696687657216287, 16.736492310713541022796947710983, 18.52908198190719665673770243049, 18.90355210094070775206291806664, 19.95318707999579105018669883366, 20.37152273181071508707127758948, 21.392998513196175168953992141127, 22.37707495495467621652809094674, 23.0577484362751684630892426010, 24.38718526700231404557003593532, 25.73717527346895154653906362654

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