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Properties

Label 1-3871-3871.2741-r0-0-0
Degree $1$
Conductor $3871$
Sign $0.474 + 0.880i$
Analytic cond. $17.9768$
Root an. cond. $17.9768$
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  + (−0.222 + 0.974i)2-s + (0.365 + 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.365 − 0.930i)10-s + (−0.222 + 0.974i)11-s + (0.0747 − 0.997i)12-s + (0.955 − 0.294i)13-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + (−0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (0.365 + 0.930i)3-s + (−0.900 − 0.433i)4-s + (−0.988 − 0.149i)5-s + (−0.988 + 0.149i)6-s + (0.623 − 0.781i)8-s + (−0.733 + 0.680i)9-s + (0.365 − 0.930i)10-s + (−0.222 + 0.974i)11-s + (0.0747 − 0.997i)12-s + (0.955 − 0.294i)13-s + (−0.222 − 0.974i)15-s + (0.623 + 0.781i)16-s + (0.826 + 0.563i)17-s + (−0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(17.9768\)
Root analytic conductor: \(17.9768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (2741, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3871,\ (0:\ ),\ 0.474 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8996866101 + 0.5371892991i\)
\(L(\frac12)\) \(\approx\) \(0.8996866101 + 0.5371892991i\)
\(L(1)\) \(\approx\) \(0.6465612382 + 0.5147358775i\)
\(L(1)\) \(\approx\) \(0.6465612382 + 0.5147358775i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T \)
3 \( 1 + (0.365 + 0.930i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (-0.222 + 0.974i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (0.826 + 0.563i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + (0.623 - 0.781i)T \)
43 \( 1 + (0.365 - 0.930i)T \)
47 \( 1 + (0.955 - 0.294i)T \)
53 \( 1 + (-0.900 - 0.433i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.222 - 0.974i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + T \)
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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

−18.706518809665113270876744034440, −18.202233764379567283045855522, −17.25639660665998433526252309480, −16.3955033493607775560822846374, −15.93031174278691861870546416204, −14.49960140570910182761687173567, −14.35752367390669381397834794932, −13.47297010341308585016998838721, −12.79213394420464348422000811304, −12.15220230345117133867545740862, −11.671931909216426735914565661588, −10.89234367152061085414831443878, −10.41016307929049085000866408071, −9.135388449580526341757028962313, −8.64075033289293893980709780306, −8.039640018592909229290755503788, −7.52374188100368648784064848155, −6.485603979438094395751453702868, −5.73196068515281091336347161310, −4.578334097003924781828606756207, −3.75685365385343170025941198925, −3.1145389321359361975757288896, −2.58849384210242949695664084390, −1.31347635298049932311630806791, −0.85829422424301058827739585027, 0.43007355759994247012894149081, 1.73310795046519726572409382512, 3.094385983759168878356585525696, 3.825461156480453753661334400606, 4.41595615542481442918780737597, 5.07627691231570271224913868923, 5.85144659854793578335781254130, 6.77502440517001949871296071380, 7.78211766605508637914006898391, 7.971696242907878584009456605786, 8.903054937296719128440370742994, 9.35383944544228031341367629550, 10.37177548979308494073185364962, 10.69442574688587988124738054209, 11.759616114920726353647994941922, 12.562847865805389120507726662835, 13.47939996179801278654370668994, 14.012730389449078994170486060661, 15.05974942972766825974478382766, 15.35032974669886887317713009417, 15.69090239531790527033233099758, 16.43605068727636579716385869497, 17.31794116075693495637471730655, 17.5707814944272980325827688227, 18.87680502507917745112328034439

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