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Properties

Label 1-221-221.129-r1-0-0
Degree $1$
Conductor $221$
Sign $0.151 - 0.988i$
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  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)15-s − 16-s − 18-s + (0.707 + 0.707i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.382 + 0.923i)3-s + i·4-s + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)6-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)10-s + (0.382 − 0.923i)11-s + (−0.923 + 0.382i)12-s + (−0.382 − 0.923i)14-s + (−0.707 − 0.707i)15-s − 16-s − 18-s + (0.707 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3805847569 + 0.3266587046i\)
\(L(\frac12)\) \(\approx\) \(-0.3805847569 + 0.3266587046i\)
\(L(1)\) \(\approx\) \(0.6966348749 + 0.7717099066i\)
\(L(1)\) \(\approx\) \(0.6966348749 + 0.7717099066i\)

Euler product

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

−25.255057248678288295220407017444, −24.4669662329831198219939486297, −23.529110724162568994613454840091, −22.87930960478060876080410491040, −22.00575498533396157579446733287, −20.420338634768257530957347642180, −20.04271269541247051703327649286, −19.13592815208402465222191247796, −18.52157396273485970212741623983, −17.08548805366301942095609584557, −15.49800758662597713525276818605, −15.05494137363928875041648047550, −13.59556581128619250805894115500, −12.95391413233133036351706714563, −12.01799249588989954124768603984, −11.52435657789371702455540059226, −9.762003635244284398499901428635, −8.93725474496619922690153142628, −7.44266870018117299594052750867, −6.535077998718087525634556659999, −5.1983838617564879640476395777, −3.79940971387984912556326462313, −2.88742112470933994683895566841, −1.522681087591459910214771644121, −0.12232211011970526320303700209, 3.12098843881172513039247813030, 3.51361606064005235920429080535, 4.6062038857015855325322964705, 5.940146918930545938255771018199, 7.07061335543521293986085750337, 8.17129602568995235187810086990, 9.11582257358667070704854191487, 10.52361634595473814673644694917, 11.52526549600070174892151647385, 12.68791001586656783091631670094, 13.94811522134902186198091356089, 14.568703362928664148330743622375, 15.67549347853056310609048848347, 16.24535980447390261124091025453, 16.90855561191996221656375938148, 18.63643772808936014686186829680, 19.6823876177193982582607858255, 20.54420570350571039860090250075, 21.66158272163544439944481269299, 22.56410539820273591152870264090, 22.94973721940371168279577013505, 24.1876479695392721707656719173, 25.17115649550171508716317698806, 26.23060536911430864910536939387, 26.72344456907824968248954318527

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