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Properties

Label 1-277-277.141-r0-0-0
Degree $1$
Conductor $277$
Sign $0.912 - 0.408i$
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  + (0.0682 + 0.997i)2-s + (−0.949 − 0.313i)3-s + (−0.990 + 0.136i)4-s + (−0.983 + 0.181i)5-s + (0.247 − 0.968i)6-s + (−0.648 + 0.761i)7-s + (−0.203 − 0.979i)8-s + (0.803 + 0.595i)9-s + (−0.247 − 0.968i)10-s + (−0.291 + 0.956i)11-s + (0.983 + 0.181i)12-s + (−0.917 − 0.398i)13-s + (−0.803 − 0.595i)14-s + (0.990 + 0.136i)15-s + (0.962 − 0.269i)16-s + (−0.995 − 0.0909i)17-s + ⋯
L(s)  = 1  + (0.0682 + 0.997i)2-s + (−0.949 − 0.313i)3-s + (−0.990 + 0.136i)4-s + (−0.983 + 0.181i)5-s + (0.247 − 0.968i)6-s + (−0.648 + 0.761i)7-s + (−0.203 − 0.979i)8-s + (0.803 + 0.595i)9-s + (−0.247 − 0.968i)10-s + (−0.291 + 0.956i)11-s + (0.983 + 0.181i)12-s + (−0.917 − 0.398i)13-s + (−0.803 − 0.595i)14-s + (0.990 + 0.136i)15-s + (0.962 − 0.269i)16-s + (−0.995 − 0.0909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(277\)
Sign: $0.912 - 0.408i$
Analytic conductor: \(1.28638\)
Root analytic conductor: \(1.28638\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{277} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 277,\ (0:\ ),\ 0.912 - 0.408i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2555592593 - 0.05462181769i\)
\(L(\frac12)\) \(\approx\) \(0.2555592593 - 0.05462181769i\)
\(L(1)\) \(\approx\) \(0.4309637891 + 0.1987481457i\)
\(L(1)\) \(\approx\) \(0.4309637891 + 0.1987481457i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad277 \( 1 \)
good2 \( 1 + (0.0682 + 0.997i)T \)
3 \( 1 + (-0.949 - 0.313i)T \)
5 \( 1 + (-0.983 + 0.181i)T \)
7 \( 1 + (-0.648 + 0.761i)T \)
11 \( 1 + (-0.291 + 0.956i)T \)
13 \( 1 + (-0.917 - 0.398i)T \)
17 \( 1 + (-0.995 - 0.0909i)T \)
19 \( 1 + (-0.334 - 0.942i)T \)
23 \( 1 + (0.983 - 0.181i)T \)
29 \( 1 + (0.613 - 0.789i)T \)
31 \( 1 + (0.648 + 0.761i)T \)
37 \( 1 + (0.990 - 0.136i)T \)
41 \( 1 + (-0.576 + 0.816i)T \)
43 \( 1 + (0.829 - 0.557i)T \)
47 \( 1 + (-0.419 - 0.907i)T \)
53 \( 1 + (-0.934 - 0.356i)T \)
59 \( 1 + (0.682 - 0.730i)T \)
61 \( 1 + (-0.682 + 0.730i)T \)
67 \( 1 + (-0.715 - 0.699i)T \)
71 \( 1 + (0.746 - 0.665i)T \)
73 \( 1 + (-0.962 - 0.269i)T \)
79 \( 1 + (-0.648 - 0.761i)T \)
83 \( 1 + (-0.247 + 0.968i)T \)
89 \( 1 + (0.538 + 0.842i)T \)
97 \( 1 + (0.949 + 0.313i)T \)
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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

−26.28986245919530269048673294015, −24.2886603099783164274393694027, −23.62412943743011860949802049968, −22.85983266046234211871438552755, −22.16850052651057439302381290629, −21.2278808458510812076834151787, −20.25919007056880514477329549390, −19.29673796137394384807465130596, −18.747406668958306828663303488534, −17.36563602615268677145146811149, −16.661257251009623029882634400737, −15.736930057502773956041251863706, −14.48739078544510945391985112774, −13.15526334164173488661373882857, −12.472900008566007867998448453828, −11.44928354352606359154319769848, −10.846471186834741568041828325928, −9.92451838193995300328951829065, −8.80977058919173589884452962765, −7.460434411521217446044152134148, −6.17341007414884532926448148754, −4.77052181184593676946881963646, −4.097222451064152427729725495182, −3.01625977077985574799183114656, −0.98273681163379771528267640234, 0.25587165280408363018141141791, 2.715577842506127971714190050612, 4.48579618576255310413857772449, 5.05066006929140621454299286058, 6.49159983120702150355614444145, 7.020039843790691270794448964745, 8.04160282837410328023665813743, 9.28752724125050460136478068144, 10.42460691706116590017215687080, 11.76112014146197552775932504664, 12.57858265098789366097303322751, 13.239307378239003851792815997216, 15.015296802400298210763949623915, 15.42296606091168309976892100811, 16.24316101794791141931428508755, 17.35114660265535816418103745194, 18.00313604251906589672071275721, 19.0090766133829514463327781992, 19.73076980989481736827119717630, 21.58442969440183766907119878689, 22.43324217254989133415840161950, 22.9284878782079286933619777659, 23.72627565220181444290700476967, 24.636806577176938491314070925, 25.3407769660207679854895733885

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