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Properties

Label 1-297-297.59-r1-0-0
Degree $1$
Conductor $297$
Sign $0.947 + 0.320i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (−0.990 + 0.139i)2-s + (0.961 − 0.275i)4-s + (0.374 + 0.927i)5-s + (0.559 − 0.829i)7-s + (−0.913 + 0.406i)8-s + (−0.5 − 0.866i)10-s + (−0.882 + 0.469i)13-s + (−0.438 + 0.898i)14-s + (0.848 − 0.529i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.939 + 0.342i)23-s + (−0.719 + 0.694i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (−0.990 + 0.139i)2-s + (0.961 − 0.275i)4-s + (0.374 + 0.927i)5-s + (0.559 − 0.829i)7-s + (−0.913 + 0.406i)8-s + (−0.5 − 0.866i)10-s + (−0.882 + 0.469i)13-s + (−0.438 + 0.898i)14-s + (0.848 − 0.529i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (0.939 + 0.342i)23-s + (−0.719 + 0.694i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.947 + 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.427747102 + 0.2350242334i\)
\(L(\frac12)\) \(\approx\) \(1.427747102 + 0.2350242334i\)
\(L(1)\) \(\approx\) \(0.8663790816 + 0.1003951290i\)
\(L(1)\) \(\approx\) \(0.8663790816 + 0.1003951290i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.990 + 0.139i)T \)
5 \( 1 + (0.374 + 0.927i)T \)
7 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (0.978 - 0.207i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (0.939 + 0.342i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
31 \( 1 + (0.0348 - 0.999i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (-0.438 + 0.898i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (-0.961 - 0.275i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (0.0348 + 0.999i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.978 - 0.207i)T \)
73 \( 1 + (-0.104 - 0.994i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (0.882 + 0.469i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−24.95004482014078224833016754873, −24.74145313343757802391682510872, −23.6827765056190080014372952820, −22.125252530276809545559864475145, −21.24914349534620766261506036672, −20.58998800901287598426988052983, −19.70368034857437604925166572411, −18.67351205764415929341818570936, −17.8616651118933068016466810282, −17.03019185417777040028055261997, −16.26770950260955959998653758149, −15.23613262106528316989111749126, −14.25058223113465044733077136302, −12.52381884934266066896253037036, −12.26573687018402041468264019896, −10.9998516121390051360606120537, −9.861955863447414737971888762846, −9.09792747611711601313339545123, −8.22982843381031781545245562082, −7.327491784740044436448041671693, −5.78529046573240277703261547202, −5.03335301415157015603311963060, −3.15039093072990561823059669617, −1.9202271940817942592309001150, −0.86684974901412937950062052120, 0.86424259226114871191539004045, 2.17168733664419236670614214004, 3.32101107052146765946162558127, 5.04977700466073844733503392340, 6.34513142058193420137040964232, 7.34889943024451931106911185524, 7.84688422557679258374144003495, 9.54033752584710696760483199088, 9.94753227866798539981655740417, 11.16693592392809959890010793671, 11.661110349397004408805762390624, 13.37767911188879116956256305633, 14.4962961854211564074042998272, 15.00943129582731967754482105753, 16.40774019590358974411246938157, 17.16810460924600136411025485189, 17.90688280169292671568200058693, 18.8185125233558334412273993083, 19.56988333631610453009066689011, 20.65317709645867338684309715202, 21.39221835745471038620856964410, 22.587092561055180662659196522449, 23.61800594729866253984201896828, 24.51030196761804191646128761713, 25.39830006220394719527772102445

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