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Properties

Label 1-221-221.178-r0-0-0
Degree $1$
Conductor $221$
Sign $-0.754 + 0.656i$
Analytic cond. $1.02631$
Root an. cond. $1.02631$
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  + (−0.866 + 0.5i)2-s + (−0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (0.707 + 0.707i)5-s + (−0.258 − 0.965i)6-s + (0.258 + 0.965i)7-s + i·8-s + (−0.866 − 0.5i)9-s + (−0.965 − 0.258i)10-s + (0.965 + 0.258i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (−0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (0.707 + 0.707i)5-s + (−0.258 − 0.965i)6-s + (0.258 + 0.965i)7-s + i·8-s + (−0.866 − 0.5i)9-s + (−0.965 − 0.258i)10-s + (0.965 + 0.258i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + 18-s + (0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(221\)    =    \(13 \cdot 17\)
Sign: $-0.754 + 0.656i$
Analytic conductor: \(1.02631\)
Root analytic conductor: \(1.02631\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{221} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 221,\ (0:\ ),\ -0.754 + 0.656i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2996473897 + 0.8011185604i\)
\(L(\frac12)\) \(\approx\) \(0.2996473897 + 0.8011185604i\)
\(L(1)\) \(\approx\) \(0.5881412169 + 0.5402502819i\)
\(L(1)\) \(\approx\) \(0.5881412169 + 0.5402502819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (0.965 + 0.258i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.965 + 0.258i)T \)
29 \( 1 + (0.258 - 0.965i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.258 + 0.965i)T \)
41 \( 1 + (-0.965 - 0.258i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.965 - 0.258i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
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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

−26.1705993011725731509373277123, −25.07833087068947549244081615924, −24.61853344797890222141970790942, −23.55424449467454415621971814341, −22.27989264460916585001307017028, −21.27734818634580946150578092127, −20.0361605742982663570324162312, −19.81327266107177046546674027718, −18.438815900224730201609067311855, −17.6516525338272793788075990159, −16.945474205694752787968136661023, −16.339058549158481163009845925726, −14.28901089195497405621195803277, −13.37948653447295470900007202438, −12.53684123016191287731277846291, −11.51854696867329507542733480397, −10.61411766177713779722767920658, −9.32250784644293453492642174112, −8.49430471077274887629087330239, −7.28821081373196712420441690463, −6.51220923656493615834323784552, −4.95525991854501238593007788621, −3.25675778436377984651806961649, −1.66199420222675109989530054947, −0.965469774738972061689834049775, 1.7875679331951910491950664721, 3.19122240320025457616203048575, 5.05076794896585598546803164410, 5.90423148891091344581521190450, 6.83315564965768397846959321795, 8.384451456907083150600881449835, 9.443639400238894485295231117490, 9.897488673371553722178605804683, 11.15361865640319662126797803792, 11.834481444764358286083231853549, 13.90674938958304197740903071260, 14.9062592295639456331528382597, 15.29778229976560019081736395643, 16.60062249802810757673968774952, 17.338039755773753533089312092194, 18.19329834793107692656302360910, 19.07872789164054835157269941997, 20.33866455918226386412974809831, 21.26555327377080962551081478715, 22.2422499472944302124138880321, 22.94197987909300188642709766407, 24.449635505059151478254942485644, 25.29550904029332912988022497709, 25.90812425035798620338209660259, 27.05321677227675677456903030291

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