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Properties

Label 1-163-163.4-r0-0-0
Degree $1$
Conductor $163$
Sign $0.412 + 0.911i$
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  + (0.996 + 0.0774i)2-s + (0.0193 + 0.999i)3-s + (0.987 + 0.154i)4-s + (0.396 + 0.918i)5-s + (−0.0581 + 0.998i)6-s + (0.813 − 0.581i)7-s + (0.973 + 0.230i)8-s + (−0.999 + 0.0387i)9-s + (0.323 + 0.946i)10-s + (−0.875 − 0.483i)11-s + (−0.135 + 0.990i)12-s + (−0.686 − 0.727i)13-s + (0.856 − 0.516i)14-s + (−0.910 + 0.413i)15-s + (0.952 + 0.305i)16-s + (−0.286 − 0.957i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0774i)2-s + (0.0193 + 0.999i)3-s + (0.987 + 0.154i)4-s + (0.396 + 0.918i)5-s + (−0.0581 + 0.998i)6-s + (0.813 − 0.581i)7-s + (0.973 + 0.230i)8-s + (−0.999 + 0.0387i)9-s + (0.323 + 0.946i)10-s + (−0.875 − 0.483i)11-s + (−0.135 + 0.990i)12-s + (−0.686 − 0.727i)13-s + (0.856 − 0.516i)14-s + (−0.910 + 0.413i)15-s + (0.952 + 0.305i)16-s + (−0.286 − 0.957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $0.412 + 0.911i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ 0.412 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.790490073 + 1.155152533i\)
\(L(\frac12)\) \(\approx\) \(1.790490073 + 1.155152533i\)
\(L(1)\) \(\approx\) \(1.745416504 + 0.7263594451i\)
\(L(1)\) \(\approx\) \(1.745416504 + 0.7263594451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (0.996 + 0.0774i)T \)
3 \( 1 + (0.0193 + 0.999i)T \)
5 \( 1 + (0.396 + 0.918i)T \)
7 \( 1 + (0.813 - 0.581i)T \)
11 \( 1 + (-0.875 - 0.483i)T \)
13 \( 1 + (-0.686 - 0.727i)T \)
17 \( 1 + (-0.286 - 0.957i)T \)
19 \( 1 + (-0.963 - 0.268i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.431 + 0.902i)T \)
31 \( 1 + (0.597 - 0.802i)T \)
37 \( 1 + (-0.835 + 0.549i)T \)
41 \( 1 + (0.987 - 0.154i)T \)
43 \( 1 + (-0.981 + 0.192i)T \)
47 \( 1 + (-0.565 + 0.824i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.973 - 0.230i)T \)
67 \( 1 + (-0.360 + 0.932i)T \)
71 \( 1 + (0.657 + 0.753i)T \)
73 \( 1 + (-0.360 - 0.932i)T \)
79 \( 1 + (0.466 + 0.884i)T \)
83 \( 1 + (0.0968 + 0.995i)T \)
89 \( 1 + (-0.875 + 0.483i)T \)
97 \( 1 + (0.533 - 0.845i)T \)
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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

−28.219222028006832598066113862649, −26.21566955039800709479573747861, −25.09557283199392787593623002816, −24.51049994548084577302313628290, −23.828022925862664471135385862615, −22.97647041706286921923069589422, −21.469085197168719517840373560995, −21.00061168434105290335817307509, −19.85146371165311540365196520061, −18.88523008436828770373688698955, −17.545358253740542018244998095184, −16.76378312076001095287539454299, −15.26359732681986542040543704156, −14.39846859205998481528897753786, −13.27585409266910107895901476606, −12.55209386783349087353166106911, −11.85988446432015285777478511761, −10.54798313267110107685334098062, −8.75733524053060659826330050259, −7.76294891701525243382104159131, −6.45569775576119618171150690242, −5.38055044629794814285818568396, −4.4892550564209238857751359120, −2.38372934449760732581283308643, −1.72674823216000576641945932964, 2.4445151282043407278024921007, 3.3638357163755212794120678127, 4.76378809193096941548820098504, 5.50123981932042100156021580201, 6.95210740330704103436163254665, 8.09209573197297952137339662064, 9.92766484757736577517331684467, 10.868972019027038399109998686217, 11.36019221578774480980860723714, 13.1413667308536363723580191051, 14.08897325151027322300645595130, 14.89026177960830289945125167270, 15.60100692932307498335701559460, 16.86811807185447809994590170751, 17.76589717278427244657662953893, 19.38766698387626989424621356262, 20.61534828432503218787983168482, 21.17482928532419117383778945318, 22.093595057214761432962759221619, 22.85516839249067082613151348856, 23.79019300030578596211874254308, 25.01778114260967297636208431644, 26.02511689465908783152721450478, 26.788454809917860207016548801134, 27.77549763569177269150152419207

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