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Properties

Label 1-43-43.15-r0-0-0
Degree $1$
Conductor $43$
Sign $-0.974 + 0.224i$
Analytic cond. $0.199691$
Root an. cond. $0.199691$
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  + (−0.222 − 0.974i)2-s + (−0.733 − 0.680i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 + 0.930i)10-s + (−0.900 − 0.433i)11-s + (0.955 + 0.294i)12-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (0.826 + 0.563i)15-s + (0.623 − 0.781i)16-s + (−0.988 − 0.149i)17-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.733 − 0.680i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 + 0.930i)10-s + (−0.900 − 0.433i)11-s + (0.955 + 0.294i)12-s + (0.365 − 0.930i)13-s + (−0.733 + 0.680i)14-s + (0.826 + 0.563i)15-s + (0.623 − 0.781i)16-s + (−0.988 − 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.974 + 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(43\)
Sign: $-0.974 + 0.224i$
Analytic conductor: \(0.199691\)
Root analytic conductor: \(0.199691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{43} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 43,\ (0:\ ),\ -0.974 + 0.224i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03489669618 - 0.3066401376i\)
\(L(\frac12)\) \(\approx\) \(-0.03489669618 - 0.3066401376i\)
\(L(1)\) \(\approx\) \(0.3069948015 - 0.3691771416i\)
\(L(1)\) \(\approx\) \(0.3069948015 - 0.3691771416i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (-0.988 + 0.149i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.988 - 0.149i)T \)
19 \( 1 + (0.0747 - 0.997i)T \)
23 \( 1 + (0.826 - 0.563i)T \)
29 \( 1 + (-0.733 + 0.680i)T \)
31 \( 1 + (0.955 + 0.294i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.900 + 0.433i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (0.826 + 0.563i)T \)
73 \( 1 + (0.365 - 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.733 - 0.680i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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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

−35.20216715465300766704862070626, −34.056154859023755683525699800480, −33.27126773490391371408950117183, −31.77981033617351490970883782818, −31.282189756132228631398801866146, −28.723211101021416582206269032386, −28.16061115118675539547319554939, −26.92165968636260530587315280252, −26.06376796851191892237217302748, −24.467294216428494220755867030340, −23.29318266455219131472296751558, −22.5944303719705488862379795798, −21.129479441409714658083579623080, −19.19917715700349906741026053283, −18.11312838127546948339321421250, −16.6094291134842821002334728310, −15.73089390545477927478650708991, −15.03844779621102605322336903782, −12.88828436974980551642735455272, −11.42788122230091777212412363674, −9.771338142306875066036435570211, −8.497237356435967123512938854303, −6.78914749528408869797025572592, −5.369142871812428125291054494866, −4.034754976978269690880841748404, 0.529152786233597396561026093606, 3.0485822611409087390100742034, 4.82148810372987725658039663499, 7.05881320317048283816271504797, 8.34357261680893085057755851413, 10.56912856909675178424211327165, 11.20157876611161662444454862370, 12.71265420851421804403822927401, 13.48805972583942463148234000946, 15.838543155392810162598078154329, 17.24429786828670950180094466176, 18.44000409440558201877045208917, 19.443133613421742782324470384772, 20.44747987938863179700395734040, 22.30372646986335564071857275906, 23.06168325732325853828218242497, 24.05248676612151045609989013852, 26.14564916342342734533114042328, 27.19694857740121948591766800711, 28.36549029129074765978383429226, 29.336825209211590434891155212228, 30.324461576019623745111001515582, 31.217864601348866161350787844806, 32.68074275174775144446186251651, 34.48528394990567031650749671359

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