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Properties

Label 1-277-277.106-r0-0-0
Degree $1$
Conductor $277$
Sign $0.107 - 0.994i$
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  + (0.990 − 0.136i)2-s + (0.113 − 0.993i)3-s + (0.962 − 0.269i)4-s + (0.158 − 0.987i)5-s + (−0.0227 − 0.999i)6-s + (0.934 + 0.356i)7-s + (0.917 − 0.398i)8-s + (−0.974 − 0.225i)9-s + (0.0227 − 0.999i)10-s + (−0.898 + 0.439i)11-s + (−0.158 − 0.987i)12-s + (0.682 + 0.730i)13-s + (0.974 + 0.225i)14-s + (−0.962 − 0.269i)15-s + (0.854 − 0.519i)16-s + (0.648 − 0.761i)17-s + ⋯
L(s)  = 1  + (0.990 − 0.136i)2-s + (0.113 − 0.993i)3-s + (0.962 − 0.269i)4-s + (0.158 − 0.987i)5-s + (−0.0227 − 0.999i)6-s + (0.934 + 0.356i)7-s + (0.917 − 0.398i)8-s + (−0.974 − 0.225i)9-s + (0.0227 − 0.999i)10-s + (−0.898 + 0.439i)11-s + (−0.158 − 0.987i)12-s + (0.682 + 0.730i)13-s + (0.974 + 0.225i)14-s + (−0.962 − 0.269i)15-s + (0.854 − 0.519i)16-s + (0.648 − 0.761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.107 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.867084365 - 1.675577905i\)
\(L(\frac12)\) \(\approx\) \(1.867084365 - 1.675577905i\)
\(L(1)\) \(\approx\) \(1.782450326 - 0.9505791774i\)
\(L(1)\) \(\approx\) \(1.782450326 - 0.9505791774i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (0.990 - 0.136i)T \)
3 \( 1 + (0.113 - 0.993i)T \)
5 \( 1 + (0.158 - 0.987i)T \)
7 \( 1 + (0.934 + 0.356i)T \)
11 \( 1 + (-0.898 + 0.439i)T \)
13 \( 1 + (0.682 + 0.730i)T \)
17 \( 1 + (0.648 - 0.761i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (-0.158 + 0.987i)T \)
29 \( 1 + (-0.715 + 0.699i)T \)
31 \( 1 + (-0.934 + 0.356i)T \)
37 \( 1 + (-0.962 + 0.269i)T \)
41 \( 1 + (-0.334 - 0.942i)T \)
43 \( 1 + (-0.613 - 0.789i)T \)
47 \( 1 + (0.983 + 0.181i)T \)
53 \( 1 + (0.949 - 0.313i)T \)
59 \( 1 + (-0.0682 - 0.997i)T \)
61 \( 1 + (0.0682 + 0.997i)T \)
67 \( 1 + (-0.877 - 0.480i)T \)
71 \( 1 + (0.803 + 0.595i)T \)
73 \( 1 + (-0.854 - 0.519i)T \)
79 \( 1 + (0.934 - 0.356i)T \)
83 \( 1 + (0.0227 + 0.999i)T \)
89 \( 1 + (0.995 - 0.0909i)T \)
97 \( 1 + (-0.113 + 0.993i)T \)
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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

−26.08696840816827523435064446503, −24.98465566007502200414636829447, −23.725787883756887298769103772636, −23.1094506150783061277789555119, −22.174191449699605581569029934, −21.33723755805952209541970887475, −20.86858968058106144294895965523, −19.87813329183492643196111692380, −18.54507751200092547303331385111, −17.35587559286237000514792970010, −16.43565329011658126240796620260, −15.23958264745481295281938319482, −14.91380684138062445935967540734, −13.96660174141035440056511215399, −13.08915985619761269521010148806, −11.52841635472142049872644498685, −10.68087687264058156773903751769, −10.41667257012853706318305845518, −8.43398916518628442637076037188, −7.61206238132059138954053575231, −6.133931226050900958262469753996, −5.35171672464056516104630901991, −4.17595836182905517439950081573, −3.27494119459527625569128527906, −2.22107239906454405466132116911, 1.50050231132447972925351351921, 2.110049167445458178132968528934, 3.72869155807745828990815387654, 5.16910768583301541457389676505, 5.62526855928553573820276031705, 7.0897756265675959860583913874, 7.97930234360341466049368422391, 9.03020062673446005295745754354, 10.68051789169629118614514072394, 11.83671351524996906867550579864, 12.329436656359124163591229335828, 13.341838192537744304337994217953, 13.98172980108710804723291298109, 15.00404226620096437666610425904, 16.13223042896383238554432396208, 17.1125011400798715974148206251, 18.25278563433526209954349092490, 19.09777522745824800472820912920, 20.43832716392751649623803737458, 20.74083402168348982429370818532, 21.66579986055319100017650319978, 23.11754979615713153992768640044, 23.78369041141283356611916194778, 24.18414408115677474422487394351, 25.38056143145622359309071411487

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