↑

Properties

Label 1-221-221.160-r1-0-0
Degree $1$
Conductor $221$
Sign $-0.673 - 0.739i$
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

Related objects

Downloads

Learn more

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.673 - 0.739i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ -0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4776949901 - 1.081492284i\)
\(L(\frac12)\) \(\approx\) \(0.4776949901 - 1.081492284i\)
\(L(1)\) \(\approx\) \(0.8404777303 - 0.4065469004i\)
\(L(1)\) \(\approx\) \(0.8404777303 - 0.4065469004i\)

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 \( 1 + (0.130 - 0.991i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.29956061245538311523417259682, −25.34358089041324699027698688847, −24.98032254575128649839310647092, −23.75144559794964602973949890519, −23.0105296568710132269118367128, −22.26554994837726941498932058931, −21.51274188502946992753262766174, −19.7896783502059751652963808719, −18.76281348673735188868860464852, −17.88206943736420382137428965979, −16.887804493474722145906231207579, −16.682778656791926239450890601, −15.01732408838054243092163415122, −14.0601287353691973677742643159, −13.206264702455562989844864631193, −12.600308076335226253226270658699, −11.098695660159137736985975019413, −9.850084917281508947574691869341, −8.78095667990824040439107911250, −7.22720901783970151389629136086, −6.64731576533378589963300675418, −5.93100658718829081247903830704, −4.657423245097742861866182940161, −3.097876131243724461631368576557, −1.266963158646853132484338199955, 0.42252541689404388839504288904, 2.01503712177827922214086178684, 3.41260567785797096738831524755, 4.50572312474836096014629561801, 5.691760966371564712261478804264, 6.30868525391845993021310202996, 8.97230971231600429855297652096, 9.31121702733307264693862240710, 10.30247917080553230505726373001, 11.3000968383456561547443150950, 12.36112793064608632949850157848, 13.07668724031440892443865263, 14.36703556792379073055918316188, 15.270337104506737492147066235827, 16.71347735912923506244358043986, 17.28967808365502481697361443270, 18.48870773468636413931057652946, 19.44744387685452620149836311714, 20.6503706633846710401192333757, 21.2784548407656721075079486888, 22.22145764345634084616752153989, 22.54933121244376691820456673909, 23.8089258495196193180475571205, 25.05100658148435095966190786196, 26.06670006612742210168946469775

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