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Properties

Label 1-221-221.127-r0-0-0
Degree $1$
Conductor $221$
Sign $0.553 - 0.832i$
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  + (−0.866 − 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (0.965 + 0.258i)7-s − i·8-s + (0.866 − 0.5i)9-s + (0.258 + 0.965i)10-s + (0.258 + 0.965i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s − 18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.965 − 0.258i)3-s + (0.5 + 0.866i)4-s + (−0.707 − 0.707i)5-s + (−0.965 − 0.258i)6-s + (0.965 + 0.258i)7-s − i·8-s + (0.866 − 0.5i)9-s + (0.258 + 0.965i)10-s + (0.258 + 0.965i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s − 18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(1.02631\)
Root analytic conductor: \(1.02631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (0:\ ),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9918971470 - 0.5316567952i\)
\(L(\frac12)\) \(\approx\) \(0.9918971470 - 0.5316567952i\)
\(L(1)\) \(\approx\) \(0.9478228524 - 0.3346255046i\)
\(L(1)\) \(\approx\) \(0.9478228524 - 0.3346255046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 + (0.965 + 0.258i)T \)
11 \( 1 + (0.258 + 0.965i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.258 - 0.965i)T \)
29 \( 1 + (-0.965 + 0.258i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (-0.965 + 0.258i)T \)
41 \( 1 + (-0.258 - 0.965i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.965 - 0.258i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.258 - 0.965i)T \)
73 \( 1 + (-0.707 - 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.258 + 0.965i)T \)
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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

−26.57360782350070992922423953152, −26.1018214846454486936129256768, −24.75421613844714932348061337053, −24.28413639304023876254598814337, −23.21059874630240560006820792620, −21.849555665114004145644725206795, −20.70547790367913060642669118684, −19.91940966454532122309342861759, −18.987080592791605252519435965123, −18.421484325831915485778364039946, −17.179370397056354002073382389837, −16.00714508503081547298837267149, −15.2920701529639020307589731194, −14.36123904365206505130278858889, −13.79802828204849033315874263266, −11.67588558606648522947361664001, −10.92426181760678034166767513519, −9.87862428610017448739999841358, −8.73240901373876244570452937646, −7.83632547973480243310925146952, −7.31945033695392142194279431560, −5.76053979200079155781709554661, −4.19162035330040588293150555657, −2.93543776723958767350219866895, −1.4579377002391952715495454933, 1.24548315276240678265690920461, 2.281070033414600142487177928528, 3.702408629368082968663975831334, 4.77787878015833851743643731076, 7.05662071564713214219537311775, 7.79096839259280745701912791021, 8.68503223000474318979073607350, 9.35385547031207752003141998365, 10.68217463987587709051475790035, 12.025483364293638148557707856827, 12.430043802063859830998368637231, 13.80651643884261291240473824533, 15.10087594109988915929624608324, 15.80813136099514881187491359768, 17.12254715875968109672325299022, 18.04600232860310559416966342810, 18.918180344312701492923557037443, 19.92494890859296640543011874915, 20.48307483910221681703295074748, 21.09563126581179159894097296639, 22.46203911982661753388061705178, 24.098622403487293032173170697777, 24.56623933435529439288942492286, 25.535197212173404980265412585235, 26.49755485662093045451837955835

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