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Properties

Label 1-221-221.126-r1-0-0
Degree $1$
Conductor $221$
Sign $-0.523 + 0.852i$
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  + (−0.258 + 0.965i)2-s + (−0.608 + 0.793i)3-s + (−0.866 − 0.5i)4-s + (−0.923 + 0.382i)5-s + (−0.608 − 0.793i)6-s + (0.793 − 0.608i)7-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.130 − 0.991i)10-s + (−0.991 + 0.130i)11-s + (0.923 − 0.382i)12-s + (0.382 + 0.923i)14-s + (0.258 − 0.965i)15-s + (0.5 + 0.866i)16-s + 18-s + (0.965 − 0.258i)19-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.608 + 0.793i)3-s + (−0.866 − 0.5i)4-s + (−0.923 + 0.382i)5-s + (−0.608 − 0.793i)6-s + (0.793 − 0.608i)7-s + (0.707 − 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.130 − 0.991i)10-s + (−0.991 + 0.130i)11-s + (0.923 − 0.382i)12-s + (0.382 + 0.923i)14-s + (0.258 − 0.965i)15-s + (0.5 + 0.866i)16-s + 18-s + (0.965 − 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.523 + 0.852i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (126, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ -0.523 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4053599935 + 0.7247116925i\)
\(L(\frac12)\) \(\approx\) \(0.4053599935 + 0.7247116925i\)
\(L(1)\) \(\approx\) \(0.5139456800 + 0.3921647823i\)
\(L(1)\) \(\approx\) \(0.5139456800 + 0.3921647823i\)

Euler product

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

−26.21516397677551420397163615919, −24.773756861318199090175160882668, −23.971151478550131936236656281232, −23.11494579932455023749382904816, −22.28408843692539089771044294390, −21.08707668141333897603286642579, −20.330078676638196864970596088190, −19.130677340060470695196886379210, −18.59290311805343939486327262144, −17.75326251604057689425581473557, −16.737684635992237901717953089361, −15.59304890958327714843511884837, −14.13407571485703543720962862642, −12.9355884913974886147303482958, −12.30932497438534695125552844666, −11.35255780751805976460419568567, −10.85740444072403861051680526168, −9.17534410299256648898603531119, −8.03574332703591669366304601500, −7.5188542996265660158458409421, −5.472144530060587779653744561246, −4.7343222706916658757568028230, −3.098945774297048292914162719814, −1.77374939163726927893842485342, −0.54572369493088951459369409060, 0.75144511218907808715123670946, 3.44369582387785349158488888678, 4.64079134452126436754579336874, 5.27829612367562292396214143499, 6.80607871643649187455056804214, 7.65622954919830537507490444757, 8.680818233515473494717613372130, 10.070228540436274435196991784106, 10.81118306128469573909431929173, 11.80740823712949216603285580270, 13.36370351383114405410057426235, 14.603129252820837024622487126087, 15.30874718726258385397620526439, 16.04565614134231816973356395699, 16.97337083860282877307031919723, 17.88637926324249127360698574446, 18.693771302913230761836043820880, 20.04559809507008588323983709212, 21.04019836575447338981933675902, 22.27879291465328493237001284126, 23.12867824255195188587418060289, 23.62718470014718098988477791811, 24.52851965820076864354295545884, 26.046880934116053612710391565887, 26.71417990975474635255623291829

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