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Properties

Label 1-3381-3381.632-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.950 + 0.310i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.128 + 0.991i)2-s + (−0.966 − 0.255i)4-s + (−0.912 + 0.409i)5-s + (0.377 − 0.925i)8-s + (−0.288 − 0.957i)10-s + (0.314 − 0.949i)11-s + (0.996 + 0.0815i)13-s + (0.869 + 0.494i)16-s + (0.704 + 0.709i)17-s + (0.327 − 0.945i)19-s + (0.986 − 0.162i)20-s + (0.900 + 0.433i)22-s + (0.665 − 0.746i)25-s + (−0.209 + 0.977i)26-s + (−0.262 + 0.965i)29-s + ⋯
L(s)  = 1  + (−0.128 + 0.991i)2-s + (−0.966 − 0.255i)4-s + (−0.912 + 0.409i)5-s + (0.377 − 0.925i)8-s + (−0.288 − 0.957i)10-s + (0.314 − 0.949i)11-s + (0.996 + 0.0815i)13-s + (0.869 + 0.494i)16-s + (0.704 + 0.709i)17-s + (0.327 − 0.945i)19-s + (0.986 − 0.162i)20-s + (0.900 + 0.433i)22-s + (0.665 − 0.746i)25-s + (−0.209 + 0.977i)26-s + (−0.262 + 0.965i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.950 + 0.310i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.950 + 0.310i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.142108867 + 0.1820704325i\)
\(L(\frac12)\) \(\approx\) \(1.142108867 + 0.1820704325i\)
\(L(1)\) \(\approx\) \(0.7913674855 + 0.3130255081i\)
\(L(1)\) \(\approx\) \(0.7913674855 + 0.3130255081i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.128 + 0.991i)T \)
5 \( 1 + (-0.912 + 0.409i)T \)
11 \( 1 + (0.314 - 0.949i)T \)
13 \( 1 + (0.996 + 0.0815i)T \)
17 \( 1 + (0.704 + 0.709i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.262 + 0.965i)T \)
31 \( 1 + (0.235 - 0.971i)T \)
37 \( 1 + (-0.390 - 0.920i)T \)
41 \( 1 + (-0.101 + 0.994i)T \)
43 \( 1 + (0.377 + 0.925i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (0.440 + 0.897i)T \)
59 \( 1 + (-0.288 - 0.957i)T \)
61 \( 1 + (0.951 + 0.307i)T \)
67 \( 1 + (-0.580 - 0.814i)T \)
71 \( 1 + (0.768 - 0.639i)T \)
73 \( 1 + (-0.403 - 0.915i)T \)
79 \( 1 + (0.786 - 0.618i)T \)
83 \( 1 + (0.339 - 0.940i)T \)
89 \( 1 + (0.894 - 0.446i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−18.89127906701635560246886348288, −18.31056800443746930549223114731, −17.547793074073863593143264435803, −16.7892291120690249069153874228, −16.09591436715612914246092521286, −15.34533236214004089862687414333, −14.505359676271254452083482481097, −13.79389515587301989716725397792, −13.04046944265511797968487541381, −12.22135731941241384600592429192, −11.916376977938186670731118479849, −11.24642729032244202130265880506, −10.30923275373075256057783283490, −9.79163848403378301230575929926, −8.89962222249692946209614091238, −8.29562450295306094579882616140, −7.62833956400715324347219834704, −6.78904435829996770710729646432, −5.48919065787859984019749566992, −4.927399804553198877443205202579, −3.841961433957622736051548849017, −3.70403769158402822641859502810, −2.58368017505425686127054836519, −1.53322321096413929952121258049, −0.87628589318735617893632997044, 0.51325304551885944751909406878, 1.42013589231376606841982135399, 3.07761964395884034306089166019, 3.61554883114431137970021045189, 4.36173279427749907480736052693, 5.27711495926042009689800465048, 6.18653715004035751583395403883, 6.57549503830741494560154414409, 7.621807080706854879711485401696, 8.01840692560006263735378556019, 8.81867633541055674638947735121, 9.380825578638113380027194375762, 10.53495415972671541568043497741, 11.043576122087660023593413288706, 11.80967040809710211263643677999, 12.79990984616595269063980627424, 13.43487654407011104960644608377, 14.22079227007931824702780156879, 14.79739530811160316567502682077, 15.43626895501963056283369598274, 16.28405484871413542319176037508, 16.3996011657456108694871229393, 17.4434295826494607060712554055, 18.18786569027355077895884053682, 18.75250281154386780090202500942

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