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Properties

Label 1-63-63.20-r0-0-0
Degree $1$
Conductor $63$
Sign $-0.342 + 0.939i$
Analytic cond. $0.292570$
Root an. cond. $0.292570$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s − 10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 8-s − 10-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.5 − 0.866i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(0.292570\)
Root analytic conductor: \(0.292570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (0:\ ),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5848818461 + 0.8352978428i\)
\(L(\frac12)\) \(\approx\) \(0.5848818461 + 0.8352978428i\)
\(L(1)\) \(\approx\) \(0.8954431031 + 0.6923561364i\)
\(L(1)\) \(\approx\) \(0.8954431031 + 0.6923561364i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−31.99243774663527132819146203647, −30.95888335854153935118747164672, −29.84452061475495958720216880350, −28.76593905865934067567157657960, −27.81713209238714104521196165823, −26.934617236798999802037220843634, −25.092307931508836039654082050908, −23.82809214140261646973097719560, −23.19708728743359872847696700867, −21.57043983198953744942275975991, −20.92813980238998848858092587711, −19.53681701562349108463893438877, −18.919270000279329932992702307033, −17.1124675864824073376317424590, −15.83741552112308547666824159107, −14.35475041691370173720337726093, −13.22572724941768578506121770247, −12.03445334948218822454021350630, −11.13785890681736891820344083328, −9.488615727841845898310548465755, −8.37098968067884270714435170930, −6.14272016679861427704041497964, −4.65583930931948334369626718444, −3.45167149451680998132885510854, −1.32821262109539135258341771297, 3.05970365132883704779328236130, 4.42515541931415033394323999481, 6.137847528495801525654894715699, 7.25306972434165787364303585032, 8.43150307697441094448459361366, 10.24074269554004572977679438718, 11.85555253194053297383153143549, 13.041768853838539638123358954091, 14.60668686839869052776940890916, 15.118774609337179397613689104199, 16.49941368005613338408600373195, 17.749162918093548281792318309656, 18.825803466462598216372321256590, 20.42652069490505187112823680086, 21.84894819157507701329106616570, 22.9077363919460848194102648851, 23.476121267368988144630363374995, 25.09940188558189703269695412665, 25.77293616548944446919539962860, 27.01938744729276436366272766231, 27.905151258129569850879188385146, 29.99619637755056753644420341532, 30.50688813647278042263812596974, 31.69105817267826320402015045987, 32.77977477191635175052479928707

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