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Properties

Label 1-63-63.16-r0-0-0
Degree $1$
Conductor $63$
Sign $0.592 + 0.805i$
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 + 5-s + 8-s + (−0.5 + 0.866i)10-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + 25-s + (−0.5 − 0.866i)26-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)4-s + 5-s + 8-s + (−0.5 + 0.866i)10-s + 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + 23-s + 25-s + (−0.5 − 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.592 + 0.805i$
Analytic conductor: \(0.292570\)
Root analytic conductor: \(0.292570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (0:\ ),\ 0.592 + 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7171710840 + 0.3625137677i\)
\(L(\frac12)\) \(\approx\) \(0.7171710840 + 0.3625137677i\)
\(L(1)\) \(\approx\) \(0.8379244875 + 0.3133759179i\)
\(L(1)\) \(\approx\) \(0.8379244875 + 0.3133759179i\)

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 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
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 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
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

−32.09134217965152011768646938980, −30.82910415226818429590032029509, −29.60431010899603816874775582863, −29.21268334996917744745905048485, −27.756415087443877904083552110674, −26.968206068168425981117027885885, −25.540276702119093919862647776817, −24.820194214566040103329617573296, −22.74486303834643485784364066382, −21.94790699879680793319694978701, −20.80918457439383955825463555863, −19.8402150986012437466029997415, −18.51753856081684662710620633265, −17.50272124872854959471415760018, −16.63194994413362820895584510725, −14.586083039714685039462603669877, −13.33698052823308211865679015300, −12.24070244105722268587363133470, −10.80594822326727780372506464341, −9.708514515951531548888379021801, −8.686461192307386427843332142167, −6.92102361092146309450821706977, −5.04482459327279130369001975293, −3.179660044695541330267763160784, −1.61823011729816480489603275011, 1.815323927137525955850154845701, 4.521434042363181263721744792389, 6.05663292553839477074012870504, 7.00587111139890202004336352453, 8.83267428611754168144913616075, 9.602714660579852966276029314009, 11.058270689532387661158297818593, 13.07323471740285680643896143714, 14.22216859070808330312693840317, 15.19129758523393687024668901429, 16.92421683909599246546521774541, 17.27243260516527457390545165920, 18.71626068736256780297147619079, 19.768641098694839253719897504213, 21.48877610368606330640471113593, 22.48503253242194658991599267489, 23.98325751917814622810078118516, 24.81536634570934709820180691451, 25.85007082781836539138985861782, 26.73400642638878563934405877734, 28.070610696555864509588425229046, 28.9486766450324522839998949958, 30.23412667191290870177421505015, 31.798845492420927951044902269706, 32.864832066351855367368332985728

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