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Properties

Label 1-221-221.165-r1-0-0
Degree $1$
Conductor $221$
Sign $-0.586 + 0.809i$
Analytic cond. $23.7497$
Root an. cond. $23.7497$
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  + (0.965 + 0.258i)2-s + (−0.991 + 0.130i)3-s + (0.866 + 0.5i)4-s + (0.923 + 0.382i)5-s + (−0.991 − 0.130i)6-s + (−0.130 + 0.991i)7-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)10-s + (−0.608 + 0.793i)11-s + (−0.923 − 0.382i)12-s + (−0.382 + 0.923i)14-s + (−0.965 − 0.258i)15-s + (0.5 + 0.866i)16-s + 18-s + (−0.258 − 0.965i)19-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.991 + 0.130i)3-s + (0.866 + 0.5i)4-s + (0.923 + 0.382i)5-s + (−0.991 − 0.130i)6-s + (−0.130 + 0.991i)7-s + (0.707 + 0.707i)8-s + (0.965 − 0.258i)9-s + (0.793 + 0.608i)10-s + (−0.608 + 0.793i)11-s + (−0.923 − 0.382i)12-s + (−0.382 + 0.923i)14-s + (−0.965 − 0.258i)15-s + (0.5 + 0.866i)16-s + 18-s + (−0.258 − 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.586 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.586 + 0.809i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ -0.586 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.222430082 + 2.394611451i\)
\(L(\frac12)\) \(\approx\) \(1.222430082 + 2.394611451i\)
\(L(1)\) \(\approx\) \(1.387681352 + 0.8430225070i\)
\(L(1)\) \(\approx\) \(1.387681352 + 0.8430225070i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (-0.991 + 0.130i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.130 + 0.991i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
19 \( 1 + (-0.258 - 0.965i)T \)
23 \( 1 + (0.608 - 0.793i)T \)
29 \( 1 + (0.130 + 0.991i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.991 + 0.130i)T \)
41 \( 1 + (-0.793 - 0.608i)T \)
43 \( 1 + (0.965 - 0.258i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (0.130 - 0.991i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.608 + 0.793i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.382 - 0.923i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.793 - 0.608i)T \)
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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

−25.712858672006680102199750525666, −24.62253190127710427562134585399, −23.92080699481276681843960834567, −23.114155663242085969899641530421, −22.31801130878092167447187918194, −21.22322186139605299538927399722, −20.85663461469578602927770030179, −19.45674884613480800408964989411, −18.3962064645615798410773042562, −17.05359768785545115513103974849, −16.60066641820452945990234720853, −15.499813960010462122121308298751, −13.99515969670566658149893909367, −13.348520252026874533711683969018, −12.61306411197081373570482426269, −11.379481162374239878384781297134, −10.5350790797630668751041309757, −9.779079998535451897864343376748, −7.734351811633310217973348853928, −6.50794716186094050248930082969, −5.72234021662932445451554984596, −4.83638036443508366959838324961, −3.60227876768498031278573920573, −1.88917233228592196892358797745, −0.70571629509750392601002578330, 1.8615433455838561657107661349, 2.98054975107346014905412067108, 4.79327756329821499877721264996, 5.36862854024285757370855645750, 6.42501580287946016921189004999, 7.146539856392880931872524338268, 8.984374758471716967218429027002, 10.354051010171345931880983667284, 11.11429003806984684762858916667, 12.45942518447765548714683122601, 12.80882341950618710648685086312, 14.166675149160501714445030029655, 15.250807070280375590455513302914, 15.88863795409139708471161436106, 17.12985309909441472583968934174, 17.83791370930021169313801014927, 18.83850840080234258353176710817, 20.52357674664367286068031916841, 21.4179694460978783827691374956, 22.03191841120476867670214725914, 22.70815321482662207041723212454, 23.66963126436722421830494738723, 24.62157035612812497531865946984, 25.49973413851928526675188624550, 26.26421179947311302267653884493

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