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Properties

Label 1-203-203.179-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.367 + 0.929i$
Analytic cond. $0.942728$
Root an. cond. $0.942728$
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  + (0.733 + 0.680i)2-s + (−0.826 − 0.563i)3-s + (0.0747 + 0.997i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (−0.623 + 0.781i)8-s + (0.365 + 0.930i)9-s + (−0.0747 − 0.997i)10-s + (−0.365 + 0.930i)11-s + (0.5 − 0.866i)12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + (−0.988 + 0.149i)16-s + (0.5 + 0.866i)17-s + (−0.365 + 0.930i)18-s + (−0.826 + 0.563i)19-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (−0.826 − 0.563i)3-s + (0.0747 + 0.997i)4-s + (−0.733 − 0.680i)5-s + (−0.222 − 0.974i)6-s + (−0.623 + 0.781i)8-s + (0.365 + 0.930i)9-s + (−0.0747 − 0.997i)10-s + (−0.365 + 0.930i)11-s + (0.5 − 0.866i)12-s + (0.623 + 0.781i)13-s + (0.222 + 0.974i)15-s + (−0.988 + 0.149i)16-s + (0.5 + 0.866i)17-s + (−0.365 + 0.930i)18-s + (−0.826 + 0.563i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.367 + 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5267831441 + 0.7749375670i\)
\(L(\frac12)\) \(\approx\) \(0.5267831441 + 0.7749375670i\)
\(L(1)\) \(\approx\) \(0.8761790173 + 0.4245099541i\)
\(L(1)\) \(\approx\) \(0.8761790173 + 0.4245099541i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.733 + 0.680i)T \)
3 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.733 - 0.680i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.623 + 0.781i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
31 \( 1 + (-0.955 + 0.294i)T \)
37 \( 1 + (-0.365 - 0.930i)T \)
41 \( 1 - T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (0.988 - 0.149i)T \)
53 \( 1 + (0.955 - 0.294i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.0747 + 0.997i)T \)
67 \( 1 + (-0.988 - 0.149i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.365 - 0.930i)T \)
83 \( 1 + (-0.900 - 0.433i)T \)
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

−27.13209930172103673515334653521, −25.6606198150835922079739206259, −24.12727813390534085035778225069, −23.40468158972864647037181115058, −22.723250036600851808845235517538, −21.953834286113042728939555195102, −21.0461896283575921710763236489, −20.149407576341926090257816965557, −18.823150320483169150241661137654, −18.33353521052791516002185294114, −16.7559350363521087198188148007, −15.59705321889719519148545047923, −15.13214220338225054007949362312, −13.83009664722795913555112012883, −12.67532825211599042190777497672, −11.61426801436321931598392850370, −10.90832976073108599730890098869, −10.31400260261742992845504099550, −8.8150674775061912987678337541, −7.05869593556161102810130399421, −5.94907137956113225976149481145, −4.9724725759577233245823225270, −3.73448578207188452204101529175, −2.9145848544572567872122185537, −0.63955002401949587584145221168, 1.772707240813069402254663454081, 3.816780049751507302098243729930, 4.771958782758120507277929477539, 5.77385795152181476002914787295, 6.93928583264749894039675824990, 7.77000555129682667982766840245, 8.84210576404276602061674003335, 10.74294762775114864950676391552, 11.84047214750042098945365875731, 12.59931135514258269368246315284, 13.215188684574814141734251146084, 14.64839389993060102353689099620, 15.663835320741248953885994986253, 16.59161458300295466154346162858, 17.180175109625537105670044053458, 18.36438455993663732217553646670, 19.46315387405680495495149454079, 20.80004220594437181814331981337, 21.59078216459588410830560332765, 22.97335369934963931989336560923, 23.36393498348598294973928260960, 24.00048452584597806345934771285, 25.04508004500726885644140557416, 25.84964472834964912159274410144, 27.20914634559102058936527227095

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