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Properties

Label 1-163-163.36-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.313 + 0.949i$
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  + (−0.0581 + 0.998i)2-s + (0.396 + 0.918i)3-s + (−0.993 − 0.116i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)6-s + (0.893 − 0.448i)7-s + (0.173 − 0.984i)8-s + (−0.686 + 0.727i)9-s + (0.597 + 0.802i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.173 + 0.984i)13-s + (0.396 + 0.918i)14-s + (0.893 + 0.448i)15-s + (0.973 + 0.230i)16-s + (0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.0581 + 0.998i)2-s + (0.396 + 0.918i)3-s + (−0.993 − 0.116i)4-s + (0.766 − 0.642i)5-s + (−0.939 + 0.342i)6-s + (0.893 − 0.448i)7-s + (0.173 − 0.984i)8-s + (−0.686 + 0.727i)9-s + (0.597 + 0.802i)10-s + (0.396 + 0.918i)11-s + (−0.286 − 0.957i)12-s + (0.173 + 0.984i)13-s + (0.396 + 0.918i)14-s + (0.893 + 0.448i)15-s + (0.973 + 0.230i)16-s + (0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7804612286 + 1.079970197i\)
\(L(\frac12)\) \(\approx\) \(0.7804612286 + 1.079970197i\)
\(L(1)\) \(\approx\) \(0.9554589575 + 0.7735903639i\)
\(L(1)\) \(\approx\) \(0.9554589575 + 0.7735903639i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.0581 + 0.998i)T \)
3 \( 1 + (0.396 + 0.918i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.893 - 0.448i)T \)
11 \( 1 + (0.396 + 0.918i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-0.835 + 0.549i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.0581 - 0.998i)T \)
31 \( 1 + (0.766 - 0.642i)T \)
37 \( 1 + (-0.939 - 0.342i)T \)
41 \( 1 + (-0.993 + 0.116i)T \)
43 \( 1 + (0.597 - 0.802i)T \)
47 \( 1 + (-0.0581 + 0.998i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (0.597 - 0.802i)T \)
73 \( 1 + (-0.993 - 0.116i)T \)
79 \( 1 + (-0.686 - 0.727i)T \)
83 \( 1 + (0.893 - 0.448i)T \)
89 \( 1 + (0.396 - 0.918i)T \)
97 \( 1 + (-0.686 - 0.727i)T \)
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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

−27.62062077192686510542833443061, −26.489659732326653107920392536240, −25.59423556179033979734893979711, −24.53637731854627179122238848534, −23.55959689031868213048542861365, −22.31054435768551194269957871300, −21.50725258298683094719788793378, −20.60478716622793314958435732247, −19.4504681482503964171091509617, −18.67838282172211671249051698821, −17.84670697126598063151411431001, −17.22915085602961361299519923110, −14.882073378920463995510922266581, −14.21884739730444891005641819234, −13.27191321266136061389604246567, −12.33731145536554312542796893083, −11.176556606561953438909156583658, −10.335527706818590458437681919555, −8.73449682334692283077485294742, −8.2366505891258327132499927980, −6.4766447809565443379118302552, −5.34855715333967448953895699985, −3.44052357973921554742102505852, −2.39497113100381722841723465962, −1.35414829015662089959972247450, 1.81871555906535965472867405234, 4.15297309265454747962618423586, 4.723011596556076296839526332864, 5.902157386526164374406683644394, 7.413285736837284428728014630180, 8.561571606484999762765537783913, 9.440041722882899276908830648149, 10.22023005590821300328147635673, 11.88343233218300615882337845368, 13.587318385999239713704911057276, 14.11656519233097305280424132065, 15.074397792840225077567186596936, 16.17354701417041763484952787191, 17.0823670175710898429677419880, 17.63130583427070784616705543915, 19.12453965563820115632406307699, 20.56481087982262319368646893334, 21.13017100133912389568568955008, 22.2206397124720898841839848666, 23.29627456201364019220458420285, 24.36978924995489124024560119593, 25.28306505894129347849352532407, 25.9280951183704819977378802063, 27.01168872644168563406053130476, 27.75447678917661456846269527722

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