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Properties

Label 1-203-203.40-r0-0-0
Degree $1$
Conductor $203$
Sign $-0.507 - 0.861i$
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  + (−0.930 − 0.365i)2-s + (−0.294 + 0.955i)3-s + (0.733 + 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.826 − 0.563i)9-s + (−0.680 + 0.733i)10-s + (−0.563 − 0.826i)11-s + (−0.866 + 0.5i)12-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)15-s + (0.0747 + 0.997i)16-s + (−0.866 − 0.5i)17-s + (0.563 + 0.826i)18-s + (0.294 + 0.955i)19-s + ⋯
L(s)  = 1  + (−0.930 − 0.365i)2-s + (−0.294 + 0.955i)3-s + (0.733 + 0.680i)4-s + (0.365 − 0.930i)5-s + (0.623 − 0.781i)6-s + (−0.433 − 0.900i)8-s + (−0.826 − 0.563i)9-s + (−0.680 + 0.733i)10-s + (−0.563 − 0.826i)11-s + (−0.866 + 0.5i)12-s + (−0.900 − 0.433i)13-s + (0.781 + 0.623i)15-s + (0.0747 + 0.997i)16-s + (−0.866 − 0.5i)17-s + (0.563 + 0.826i)18-s + (0.294 + 0.955i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $-0.507 - 0.861i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ -0.507 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1769892082 - 0.3096504654i\)
\(L(\frac12)\) \(\approx\) \(0.1769892082 - 0.3096504654i\)
\(L(1)\) \(\approx\) \(0.5048161601 - 0.1120818304i\)
\(L(1)\) \(\approx\) \(0.5048161601 - 0.1120818304i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.930 - 0.365i)T \)
3 \( 1 + (-0.294 + 0.955i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (-0.563 - 0.826i)T \)
13 \( 1 + (-0.900 - 0.433i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.294 + 0.955i)T \)
23 \( 1 + (-0.988 - 0.149i)T \)
31 \( 1 + (-0.149 - 0.988i)T \)
37 \( 1 + (-0.563 + 0.826i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (0.997 - 0.0747i)T \)
53 \( 1 + (-0.988 + 0.149i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.680 - 0.733i)T \)
67 \( 1 + (-0.0747 + 0.997i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (0.930 - 0.365i)T \)
79 \( 1 + (0.563 - 0.826i)T \)
83 \( 1 + (0.222 - 0.974i)T \)
89 \( 1 + (0.930 + 0.365i)T \)
97 \( 1 + (0.974 + 0.222i)T \)
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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

−26.9063636407060110140376966056, −26.12546775839328750217368149852, −25.43926478167784921291419877550, −24.41177216971157730956145399464, −23.68020087132346643498799060718, −22.64662533351315518133345979984, −21.56256161387240876185839288537, −19.92255607846683857237023514291, −19.444465833156835206702578760916, −18.17062864093628951701091710949, −17.8954935980812107950862260716, −16.99875099668785125176176947211, −15.64242665473559808976719527894, −14.65906849481423501381709868377, −13.69295429492864707690405227962, −12.33575197036471795733887687295, −11.246053034026841690118698173509, −10.37167714352223814917954372692, −9.270891536007193605557311070861, −7.87203435949365448089932255553, −7.058886661783627279722181426604, −6.38787956615980192841240938018, −5.1066751079047067928585164948, −2.60701501557903987629083603990, −1.84325525736870673522044995126, 0.34936436232597340757333496363, 2.26006246243885249160776693018, 3.66752672577490960821266346406, 5.04372581028310032028768735349, 6.12377962430404935589055779101, 7.87652252372950473674399968225, 8.78602051014825003019442436053, 9.72888388231587865953158409876, 10.4471063696681619392393145583, 11.61472536875115927598627358204, 12.48000376230188525210305499491, 13.82440945500305839909142436785, 15.42183596811109695022818610703, 16.16500759570916402900795834150, 16.94654354316551797383792343179, 17.690556453104040475714988390065, 18.865023683444642500926197531437, 20.32725261851810154094971880124, 20.47005972436991664505462639378, 21.67479871964897457057120933199, 22.28516226999548720142771310498, 23.9643502170352044192032552546, 24.792552910565457690926072813244, 25.85696875639866560069349487842, 26.82395447019494081021611066813

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