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Properties

Label 1-221-221.108-r1-0-0
Degree $1$
Conductor $221$
Sign $-0.0882 + 0.996i$
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 + (−0.793 − 0.608i)3-s + (−0.866 − 0.5i)4-s + (0.382 + 0.923i)5-s + (0.793 − 0.608i)6-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (−0.991 + 0.130i)10-s + (−0.130 − 0.991i)11-s + (0.382 + 0.923i)12-s + (−0.923 + 0.382i)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 + (−0.793 − 0.608i)3-s + (−0.866 − 0.5i)4-s + (0.382 + 0.923i)5-s + (0.793 − 0.608i)6-s + (0.608 + 0.793i)7-s + (0.707 − 0.707i)8-s + (0.258 + 0.965i)9-s + (−0.991 + 0.130i)10-s + (−0.130 − 0.991i)11-s + (0.382 + 0.923i)12-s + (−0.923 + 0.382i)14-s + (0.258 − 0.965i)15-s + (0.5 + 0.866i)16-s − 18-s + (0.965 − 0.258i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0882 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0882 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.0882 + 0.996i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ -0.0882 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8493489725 + 0.9279438624i\)
\(L(\frac12)\) \(\approx\) \(0.8493489725 + 0.9279438624i\)
\(L(1)\) \(\approx\) \(0.7413276473 + 0.3840377649i\)
\(L(1)\) \(\approx\) \(0.7413276473 + 0.3840377649i\)

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 \( 1 + (-0.793 - 0.608i)T \)
5 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 + (0.608 + 0.793i)T \)
11 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (0.965 - 0.258i)T \)
23 \( 1 + (-0.130 - 0.991i)T \)
29 \( 1 + (0.608 - 0.793i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (0.793 + 0.608i)T \)
41 \( 1 + (-0.991 + 0.130i)T \)
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 \( 1 + (0.608 + 0.793i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.130 - 0.991i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.991 + 0.130i)T \)
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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

−26.444137492391756961055571046640, −25.224353946917045757109615238317, −23.76100061768180307050614731242, −23.1750048132300485860973017978, −22.05189831681797804188530419426, −21.19348685974946852369207653109, −20.48405607280809378816639571652, −19.88505368638980900892632342570, −18.14861324464053967722154730142, −17.54538884937019429185741043931, −16.88092881863828760725932330548, −15.81908083752102802412583389069, −14.29471977827180560566560384860, −13.23109550997982656453298477477, −12.194933212655831449067027001290, −11.488279973166704797953983018113, −10.23274967301516036783616308761, −9.77209893456777632884090769875, −8.56939331340404378268993650852, −7.25085100893697901228990164864, −5.343498677064920654125741985756, −4.700992020465243854097447424335, −3.68523030333710307486443140557, −1.719402709562805411740548005866, −0.691658196491957817483729378933, 0.994517449853217369518898684553, 2.63877441217007771754068219492, 4.718809643406799558785644644994, 5.85031454510455269502301933466, 6.35095541863893328351228693298, 7.57233199333755388933170708049, 8.44494446510677044066342187425, 9.87494546543951903782774875698, 10.94734804780893526086004651802, 11.88314499859526532174471927432, 13.371751313100344911518103119, 14.06706008974085490909284436165, 15.1477217264683133964980004057, 16.14000640323911855748534559150, 17.170315255178528486440126609655, 18.09313131988357698325131390603, 18.51663405855380973997250111197, 19.336906833326051137346629062036, 21.34496755048661595177381801936, 22.208035420699561075960445692805, 22.81300535447618281566522569530, 24.009272862347500083099162291200, 24.57420763671332633572110927071, 25.378506499277232606774979964132, 26.56469610965677602797261398440

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