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Properties

Label 1-221-221.75-r1-0-0
Degree $1$
Conductor $221$
Sign $0.278 - 0.960i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $0.278 - 0.960i$
Analytic conductor: \(23.7497\)
Root analytic conductor: \(23.7497\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (1:\ ),\ 0.278 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465577171 - 1.100534536i\)
\(L(\frac12)\) \(\approx\) \(1.465577171 - 1.100534536i\)
\(L(1)\) \(\approx\) \(1.102466194 - 0.2340824947i\)
\(L(1)\) \(\approx\) \(1.102466194 - 0.2340824947i\)

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

−26.29034451072756056192033306934, −25.48815735291837968442534691636, −25.20633076790164427479669542833, −24.158041248362948406969029933452, −22.4242404043074898683866553015, −21.32029716130139059451807148138, −20.86573229626123846742029946547, −19.78298250495565757144358427770, −18.726150704243387051277143119679, −18.262175567666784034429026091415, −17.33799256206407547753829092988, −15.8491620742969573380559805584, −15.1823843957103970227100426855, −14.04264468057291228432365228702, −12.84378553207047943789285159014, −11.99900851400136871393382352274, −10.36877260752587321778712979683, −9.74063297050068395995039911226, −8.892266635980880538755341255541, −7.857714812892515690919976487133, −6.834585061470837279209567223131, −5.57203277799090145951689504519, −3.47629327278961621212334101375, −2.35060534012320757828106671381, −1.72947154050593304651257211976, 0.69963445545040496881094646614, 1.99775069787047242307702241238, 3.18265514283913861000425244164, 4.90645268288324004123699387568, 6.33999436482271973523072015097, 7.434217128437204167374131144221, 8.427103720758770740354946235816, 9.29870452441101202502175848896, 10.17512581285761156146778194617, 10.95567393014724956672289945813, 12.834670497167701202302368152982, 13.75905851917538831694425702867, 14.52950924427168028935742679504, 15.890147887111837668518971221179, 16.52523615960713752628998952234, 17.60609985755023431692413503414, 18.52143316481282180749633380573, 19.5267390029229676359285631082, 20.41462584473686145760759019760, 20.92767225297628237702346694274, 22.083565271790308470984618140746, 24.00677575942574350011019913827, 24.25284496261043325111998173618, 25.51123665313308884800708692154, 26.10926551528069530437817064582

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