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Properties

Label 1-277-277.98-r1-0-0
Degree $1$
Conductor $277$
Sign $0.772 + 0.634i$
Analytic cond. $29.7677$
Root an. cond. $29.7677$
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  + (−0.136 − 0.990i)2-s + (0.803 + 0.595i)3-s + (−0.962 + 0.269i)4-s + (−0.356 − 0.934i)5-s + (0.480 − 0.877i)6-s + (0.158 + 0.987i)7-s + (0.398 + 0.917i)8-s + (0.291 + 0.956i)9-s + (−0.877 + 0.480i)10-s + (0.557 − 0.829i)11-s + (−0.934 − 0.356i)12-s + (−0.682 − 0.730i)13-s + (0.956 − 0.291i)14-s + (0.269 − 0.962i)15-s + (0.854 − 0.519i)16-s + (−0.181 + 0.983i)17-s + ⋯
L(s)  = 1  + (−0.136 − 0.990i)2-s + (0.803 + 0.595i)3-s + (−0.962 + 0.269i)4-s + (−0.356 − 0.934i)5-s + (0.480 − 0.877i)6-s + (0.158 + 0.987i)7-s + (0.398 + 0.917i)8-s + (0.291 + 0.956i)9-s + (−0.877 + 0.480i)10-s + (0.557 − 0.829i)11-s + (−0.934 − 0.356i)12-s + (−0.682 − 0.730i)13-s + (0.956 − 0.291i)14-s + (0.269 − 0.962i)15-s + (0.854 − 0.519i)16-s + (−0.181 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.772 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.772 + 0.634i$
Analytic conductor: \(29.7677\)
Root analytic conductor: \(29.7677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (1:\ ),\ 0.772 + 0.634i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.454451399 + 0.5204649402i\)
\(L(\frac12)\) \(\approx\) \(1.454451399 + 0.5204649402i\)
\(L(1)\) \(\approx\) \(1.066227098 - 0.1576453272i\)
\(L(1)\) \(\approx\) \(1.066227098 - 0.1576453272i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (-0.136 - 0.990i)T \)
3 \( 1 + (0.803 + 0.595i)T \)
5 \( 1 + (-0.356 - 0.934i)T \)
7 \( 1 + (0.158 + 0.987i)T \)
11 \( 1 + (0.557 - 0.829i)T \)
13 \( 1 + (-0.682 - 0.730i)T \)
17 \( 1 + (-0.181 + 0.983i)T \)
19 \( 1 + (-0.775 + 0.631i)T \)
23 \( 1 + (0.934 - 0.356i)T \)
29 \( 1 + (0.247 + 0.968i)T \)
31 \( 1 + (-0.987 - 0.158i)T \)
37 \( 1 + (0.269 + 0.962i)T \)
41 \( 1 + (0.334 + 0.942i)T \)
43 \( 1 + (0.926 + 0.377i)T \)
47 \( 1 + (0.648 - 0.761i)T \)
53 \( 1 + (-0.665 + 0.746i)T \)
59 \( 1 + (0.0682 + 0.997i)T \)
61 \( 1 + (0.997 - 0.0682i)T \)
67 \( 1 + (0.0227 + 0.999i)T \)
71 \( 1 + (0.113 - 0.993i)T \)
73 \( 1 + (-0.519 + 0.854i)T \)
79 \( 1 + (-0.158 + 0.987i)T \)
83 \( 1 + (0.877 + 0.480i)T \)
89 \( 1 + (0.419 - 0.907i)T \)
97 \( 1 + (0.595 - 0.803i)T \)
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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

−25.44761980859336023523819691389, −24.50631393230635119322117678956, −23.57697714529178318777259152314, −23.02821012261345388195881246875, −21.985495338967192454616860483381, −20.5986462704062599612181732610, −19.41263357350115793769956108531, −19.08745992981238011995385746211, −17.774925450279911907291406326371, −17.31883179668827511606641547326, −15.916457867688233721964249816289, −14.88253563678878038854658919455, −14.35846372476512218886931307742, −13.62372149901901648508935535344, −12.49375561891349352696109373771, −11.100933772360072282149935612202, −9.765496298895773295227071824698, −9.02031011712092283090509186959, −7.51903140761698916273262005926, −7.23583325489464525617182452902, −6.49253181114363175810721716232, −4.58857049573248299953488565660, −3.73233573634659115584586820531, −2.17627526648334021452157724123, −0.479332994895876162626598295, 1.31650535368978939451400780179, 2.5758871484305552545849279281, 3.638700426978912334655044914430, 4.64238300944143547589382343600, 5.62044380393351216612667573772, 7.94196716242860766364508580968, 8.6602395048086180822766166839, 9.16609505363391649575606636714, 10.36392820838433115771639898861, 11.355101207979992995362715695986, 12.54100343766386479283791033911, 13.031467426866521460051335798822, 14.451674576717111463040812511353, 15.11397813232026649913847268577, 16.42666718913399018045393676864, 17.20387147859585081035995531592, 18.647698735593933286227853713, 19.400787959819888071362782334086, 20.0333414781598920835157691868, 20.97744214189814216234689023646, 21.660473611775365409429266505887, 22.31438710041138217344974646143, 23.74848959397825328630324863470, 24.79760751028200813027628575901, 25.52106971177551680745564984735

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