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Properties

Label 1-63-63.23-r1-0-0
Degree $1$
Conductor $63$
Sign $0.235 - 0.971i$
Analytic cond. $6.77029$
Root an. cond. $6.77029$
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

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

Invariants

Degree: \(1\)
Conductor: \(63\)    =    \(3^{2} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(6.77029\)
Root analytic conductor: \(6.77029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{63} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 63,\ (1:\ ),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8033746580 - 0.6318401821i\)
\(L(\frac12)\) \(\approx\) \(0.8033746580 - 0.6318401821i\)
\(L(1)\) \(\approx\) \(0.7595209290 - 0.2230908057i\)
\(L(1)\) \(\approx\) \(0.7595209290 - 0.2230908057i\)

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

−32.61225768814906314220135269219, −30.94419651843354149861328901753, −29.67650030471259265598670916732, −29.1997344562535363172594189137, −27.70758171034358100066058176591, −26.756919962340780062744382109787, −25.857522354158559317988400654136, −24.85983357857390609664841934105, −23.5872882714411633084636646352, −21.88421847629315040628768889288, −21.097145541465031753558850614620, −19.34470498109147253267147116332, −18.84759497291656897156220030202, −17.45452506559098534899318228401, −16.61357419310751345755193203093, −15.081464827480968121213017597262, −14.00355635777928202599774433529, −12.046726385166530383194757060786, −10.84284691861019602956782030328, −9.81780931370748882162203845038, −8.50643925655536462033032817346, −7.00417138625683044129903071946, −5.9613981523580490597959675102, −3.30941747906127098323827891276, −1.6668577080388996842174064824, 0.77544598963898912276770418978, 2.4768014883637135034269801651, 4.91672152841810865616156128045, 6.56941148756834149712064149289, 8.013863138467497080827713100940, 9.27779669909701517737184372786, 10.16697128773137519542390946739, 11.815619581035008123069294182124, 12.91282259570478444132538397939, 14.745328357063747412473056532418, 16.03427127208966251922082368377, 17.20544139008241069655580871081, 17.86918037905079513043693841844, 19.45124018626284195726775719601, 20.36117290788421486883145163913, 21.31100447662073464698699199080, 22.96858190346498249574834616223, 24.665121829090576434262893038133, 25.045270140140455902089896893844, 26.34628497314741057210957012028, 27.65617351579678535918080110412, 28.30044707855615115926812573428, 29.44421870915632692720137859804, 30.3913704253199067440151370137, 32.11127961126652891510768805344

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