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Properties

Label 1-63-63.25-r0-0-0
Degree $1$
Conductor $63$
Sign $0.971 - 0.235i$
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  + 2-s + 4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 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 + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + 2-s + 4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + 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 + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.971 - 0.235i$
Analytic conductor: \(0.292570\)
Root analytic conductor: \(0.292570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (0:\ ),\ 0.971 - 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436415598 - 0.1716779885i\)
\(L(\frac12)\) \(\approx\) \(1.436415598 - 0.1716779885i\)
\(L(1)\) \(\approx\) \(1.544080465 - 0.1212792210i\)
\(L(1)\) \(\approx\) \(1.544080465 - 0.1212792210i\)

Euler product

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

−32.19656956688511423171633757470, −31.38354802722667209021170228863, −30.198307482762071458900535754864, −29.5991720505558032551659137816, −28.13207389467879360277721741227, −26.667110846739718933720825999003, −25.68221593104835352201364562805, −24.25648926244939257041347423235, −23.441559335291343922269138706, −22.24780988518929291858495626411, −21.53636394778561573028411369018, −19.988366946696056638161342711651, −19.09723212100276036939750918257, −17.46716861508634850381543071392, −15.82121162187913886014945947311, −15.08716606832238207327774198588, −13.85946696344231432630432418067, −12.67201770052863198631015700803, −11.27680424580001237602032757318, −10.45170623480493104006616456509, −8.08882544457826254956186566496, −6.82895774292570260806191718216, −5.4925545335228443821026322492, −3.806710223879517089444640946548, −2.594916183043727279383412342332, 2.1529959313093544231969574101, 4.14022906279380746878277374178, 5.04673640052099617779085922190, 6.77027576449304127125513693512, 8.1231298607054143440724777139, 9.94039552823299905250001110737, 11.64719657936340570885271950233, 12.44344646685592426687320555254, 13.62693597437755972136620955645, 14.988363826995783892640581326566, 16.03532399263962021408568402645, 17.06571045062561068492299915515, 18.987152355584578371776042771983, 20.306110030425296125436977741168, 20.92130282078276344863347994774, 22.37708662019528224026103520488, 23.39647963853528082536616738209, 24.317844932477225142751654136640, 25.27189730300615054107016496546, 26.72131737528539753035523890827, 28.28428480725796869335824495436, 28.99937466633912832789599636201, 30.38838507045460863059745361144, 31.48669907677979793529516492932, 31.97528462018729448844350280799

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