Properties

Label 1-163-163.54-r0-0-0
Degree $1$
Conductor $163$
Sign $0.434 - 0.900i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (0.713 − 0.700i)2-s + (−0.981 − 0.192i)3-s + (0.0193 − 0.999i)4-s + (0.597 + 0.802i)5-s + (−0.835 + 0.549i)6-s + (0.996 − 0.0774i)7-s + (−0.686 − 0.727i)8-s + (0.925 + 0.378i)9-s + (0.987 + 0.154i)10-s + (0.323 + 0.946i)11-s + (−0.211 + 0.977i)12-s + (−0.286 − 0.957i)13-s + (0.657 − 0.753i)14-s + (−0.431 − 0.902i)15-s + (−0.999 − 0.0387i)16-s + (0.973 − 0.230i)17-s + ⋯
L(s)  = 1  + (0.713 − 0.700i)2-s + (−0.981 − 0.192i)3-s + (0.0193 − 0.999i)4-s + (0.597 + 0.802i)5-s + (−0.835 + 0.549i)6-s + (0.996 − 0.0774i)7-s + (−0.686 − 0.727i)8-s + (0.925 + 0.378i)9-s + (0.987 + 0.154i)10-s + (0.323 + 0.946i)11-s + (−0.211 + 0.977i)12-s + (−0.286 − 0.957i)13-s + (0.657 − 0.753i)14-s + (−0.431 − 0.902i)15-s + (−0.999 − 0.0387i)16-s + (0.973 − 0.230i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.434 - 0.900i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.434 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.214906103 - 0.7628722083i\)
\(L(\frac12)\) \(\approx\) \(1.214906103 - 0.7628722083i\)
\(L(1)\) \(\approx\) \(1.216350339 - 0.5249168386i\)
\(L(1)\) \(\approx\) \(1.216350339 - 0.5249168386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.713 - 0.700i)T \)
3 \( 1 + (-0.981 - 0.192i)T \)
5 \( 1 + (0.597 + 0.802i)T \)
7 \( 1 + (0.996 - 0.0774i)T \)
11 \( 1 + (0.323 + 0.946i)T \)
13 \( 1 + (-0.286 - 0.957i)T \)
17 \( 1 + (0.973 - 0.230i)T \)
19 \( 1 + (-0.910 - 0.413i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.249 - 0.968i)T \)
31 \( 1 + (-0.993 + 0.116i)T \)
37 \( 1 + (0.893 - 0.448i)T \)
41 \( 1 + (0.0193 + 0.999i)T \)
43 \( 1 + (-0.360 + 0.932i)T \)
47 \( 1 + (-0.963 - 0.268i)T \)
53 \( 1 + (0.173 + 0.984i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.686 + 0.727i)T \)
67 \( 1 + (0.856 - 0.516i)T \)
71 \( 1 + (-0.627 - 0.778i)T \)
73 \( 1 + (0.856 + 0.516i)T \)
79 \( 1 + (-0.135 + 0.990i)T \)
83 \( 1 + (-0.565 - 0.824i)T \)
89 \( 1 + (0.323 - 0.946i)T \)
97 \( 1 + (-0.790 - 0.612i)T \)
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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

−27.65863473784472670695479465110, −27.16909030429908338077755981449, −25.71581141662798031659883634976, −24.68790165164158831625868897419, −23.892534086724676720954103845450, −23.43296685258000606957817496403, −21.77763869143682054877714720605, −21.5560161312799348379101341406, −20.6807375164201184021594977557, −18.75395497989594514789985823821, −17.53999737001147171889477607097, −16.81697296055052606399548129379, −16.330322908680571518341723622636, −14.87664481773123674742701285317, −13.97581658532112065417527916741, −12.78097396196641696672517262194, −11.908950747012135429199803489119, −10.95951974186695760059614607423, −9.2866061652516676791373155706, −8.19689114019825432119653696983, −6.76135285532161360318763388397, −5.64538163216213004439424056191, −5.00299184424993838997643432908, −3.877455007256139291036118442054, −1.61359232700660728588228523105, 1.385170769146429171423144280164, 2.6076084399903879606396694523, 4.397373245774400472296159788219, 5.32766692232949941329326403313, 6.37166025589476551180934309800, 7.515554133141304467220560011976, 9.69502987323116385545272360174, 10.56554028227387191741415315192, 11.2919825253752774195264833652, 12.368239626807446132192132806510, 13.26536808256734137349445310447, 14.63144409330014745933712549967, 15.10691735663782002155167652763, 16.93901054723238955243327930098, 17.91250777744504204657373342802, 18.51684893245857132033113600528, 19.84036805682151956245688327329, 21.12773960782128724136547541627, 21.67807118242136934107280597248, 22.92939670030739272919543592491, 23.10410248806604521902665910118, 24.504817278359489727326914469249, 25.272183198110516079880456655120, 27.065225436242159564061273636800, 27.78800111734383497024444180179

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