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Properties

Label 1-221-221.217-r0-0-0
Degree $1$
Conductor $221$
Sign $-0.115 + 0.993i$
Analytic cond. $1.02631$
Root an. cond. $1.02631$
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  + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + i·12-s + i·14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.5 + 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s − 8-s + (0.5 − 0.866i)9-s + (−0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + i·12-s + i·14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + 18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.115 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.115 + 0.993i$
Analytic conductor: \(1.02631\)
Root analytic conductor: \(1.02631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (0:\ ),\ -0.115 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.280282061 + 1.438239017i\)
\(L(\frac12)\) \(\approx\) \(1.280282061 + 1.438239017i\)
\(L(1)\) \(\approx\) \(1.394016325 + 0.8864847345i\)
\(L(1)\) \(\approx\) \(1.394016325 + 0.8864847345i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + iT \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + iT \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.866 - 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 - iT \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−26.60700702373639560777664851850, −25.17184389454975497966041819449, −24.166860265682120950106904839071, −23.647416988421216273060234011447, −22.18897915183334418718483675633, −21.241503133821454804238533501152, −20.59338866135547618464213009363, −20.18216826414223916332460516116, −19.02479764140268185257577565128, −18.00806066843955089064026244129, −16.56719453402907677491383677640, −15.60859107524199144172694372168, −14.43386686658501379771551561422, −13.736478707781224277327469691595, −12.89950632455601657824061522899, −11.72901425296229553089227668583, −10.566081163315266259816813432698, −9.77974812182433899500676493794, −8.56492789690545578548946797893, −7.88136211953418245359334378910, −5.63293756525036942332743730921, −4.67594247183675862836359657953, −3.88937028230928148155832204934, −2.51152326943936640775693870472, −1.27443352922073422691716219910, 2.22710461444014247519790670075, 3.13762842271383813815850574324, 4.53000372059129142805672681406, 5.84216831971170029132442074130, 7.06977674429355484753653103166, 7.728467638366659391196378966586, 8.643566647871889697092790762654, 9.92375728656843349316129170030, 11.5072283654404687342458166052, 12.5294941576182922390992634253, 13.713702548246395512803079970739, 14.27840004568300863858886813589, 15.26742342982911142173496169865, 15.71183979850089763117412360616, 17.76003655981172559570377827472, 17.94420859640256446425259798870, 18.98116662315117047744841545403, 20.32629382720692327806774902686, 21.367224893735411415880127367377, 22.06343642673156920853556153987, 23.433379452191837268653775513536, 23.88774970371963944599704868801, 25.075373391785299251950275475602, 25.59658918677232697547137561700, 26.54779605009325697222109663298

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