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Properties

Label 1-197-197.129-r1-0-0
Degree $1$
Conductor $197$
Sign $-0.880 - 0.474i$
Analytic cond. $21.1705$
Root an. cond. $21.1705$
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  + (−0.433 + 0.900i)2-s + (0.433 + 0.900i)3-s + (−0.623 − 0.781i)4-s + (−0.781 − 0.623i)5-s − 6-s + (0.900 − 0.433i)7-s + (0.974 − 0.222i)8-s + (−0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.433 + 0.900i)11-s + (0.433 − 0.900i)12-s + (0.974 + 0.222i)13-s + i·14-s + (0.222 − 0.974i)15-s + (−0.222 + 0.974i)16-s + (0.781 + 0.623i)17-s + ⋯
L(s)  = 1  + (−0.433 + 0.900i)2-s + (0.433 + 0.900i)3-s + (−0.623 − 0.781i)4-s + (−0.781 − 0.623i)5-s − 6-s + (0.900 − 0.433i)7-s + (0.974 − 0.222i)8-s + (−0.623 + 0.781i)9-s + (0.900 − 0.433i)10-s + (−0.433 + 0.900i)11-s + (0.433 − 0.900i)12-s + (0.974 + 0.222i)13-s + i·14-s + (0.222 − 0.974i)15-s + (−0.222 + 0.974i)16-s + (0.781 + 0.623i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.880 - 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(197\)
Sign: $-0.880 - 0.474i$
Analytic conductor: \(21.1705\)
Root analytic conductor: \(21.1705\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{197} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 197,\ (1:\ ),\ -0.880 - 0.474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1829399841 + 0.7243703506i\)
\(L(\frac12)\) \(\approx\) \(-0.1829399841 + 0.7243703506i\)
\(L(1)\) \(\approx\) \(0.5787503137 + 0.5078908072i\)
\(L(1)\) \(\approx\) \(0.5787503137 + 0.5078908072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad197 \( 1 \)
good2 \( 1 + (-0.433 + 0.900i)T \)
3 \( 1 + (0.433 + 0.900i)T \)
5 \( 1 + (-0.781 - 0.623i)T \)
7 \( 1 + (0.900 - 0.433i)T \)
11 \( 1 + (-0.433 + 0.900i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (0.781 + 0.623i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + (-0.433 + 0.900i)T \)
37 \( 1 + (-0.222 - 0.974i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (0.900 + 0.433i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (0.623 + 0.781i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + (-0.974 + 0.222i)T \)
71 \( 1 + (0.781 + 0.623i)T \)
73 \( 1 + (-0.974 + 0.222i)T \)
79 \( 1 + (-0.781 + 0.623i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−26.13706204581193727655150017881, −25.58248329111924563321690685961, −24.10905217687477702555139638185, −23.43174014075835175845791153737, −22.30547109939435434047350283431, −21.1035643148803768268241957340, −20.38672817287567765656172766269, −19.26744268658860716428670385570, −18.493020048417633126359296584243, −18.21755065205589336249054673969, −16.79673700167353673017138424074, −15.346491046100135863607451297773, −14.19071446698950876387068934543, −13.35197302812572441914111757575, −12.10369709133151962665999787684, −11.4412815154200566126955817769, −10.57234569125206967787833869831, −8.84725217580578483547067820835, −8.15951707711564241172413548649, −7.40709129024057154603795257672, −5.74254704204652472071862347650, −3.87735681701974805907584217193, −2.92313589092311052159709155241, −1.725395451652087410478453153269, −0.28919403342109525997148379660, 1.596762045000448526217572612546, 3.99419849115536594576576066585, 4.551264388498929382104105997621, 5.73738891438986923605483373705, 7.52572718861592240919975193745, 8.1981691858792372080175547435, 9.01241479951083264203121985021, 10.287085030857676986394493099194, 11.08667509258569824801161039802, 12.746461737365853970757000700196, 14.0861419971133415896633372416, 14.893134502449805842781623894904, 15.67633899670307496170153589916, 16.53494174039209908304552429072, 17.33615857818141145836444434867, 18.57247379897901518098350675987, 19.726680676188779417922631539455, 20.51853924034455349631908618975, 21.389432196945528888332055466019, 23.00011238822604294955558894453, 23.48737278126820160174215416994, 24.49713670242327359674268700760, 25.635265857088557046051053520330, 26.26545618195879504530484851508, 27.26708990441497198057662021746

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