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Properties

Label 1-203-203.23-r0-0-0
Degree $1$
Conductor $203$
Sign $0.839 + 0.543i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $0.839 + 0.543i$
Analytic conductor: \(0.942728\)
Root analytic conductor: \(0.942728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{203} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 203,\ (0:\ ),\ 0.839 + 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9624375729 + 0.2842820903i\)
\(L(\frac12)\) \(\approx\) \(0.9624375729 + 0.2842820903i\)
\(L(1)\) \(\approx\) \(0.9104326097 + 0.1858627215i\)
\(L(1)\) \(\approx\) \(0.9104326097 + 0.1858627215i\)

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

−26.87211756668725629703692864282, −26.20715738802814165142091951062, −25.001396656055231895033694185179, −24.270773833391570332579068734656, −22.715907703563336800738281629786, −21.61381145320242969328130239716, −20.851074460674231983642934358240, −20.077186850424096877448839251288, −19.296022907746710836880034449809, −18.56518909815141182697691894066, −17.01863939252785154271668117558, −16.17119150342552808484919828967, −15.54925724429762002402215834791, −13.88310850921202657429298610263, −13.15042690999806378250697891300, −11.63104825594916051305544863528, −11.134134780612742444383263637531, −9.59450223838338477071139492599, −8.917232319058497133620808736401, −8.19100796583519358720524409089, −7.02475137519721813046613724494, −4.81370947385174898526331943284, −3.79199285841237179105545585230, −2.83019834289752226315638074654, −1.15529641878833059750465880545, 1.30957164353476449156553456701, 2.81571206094011609137392615960, 4.24049857260352777645733289763, 6.10291168969101283836507355053, 7.07855927447631189656248963406, 7.834719418079235629289367599735, 8.713273941305755832125051947632, 9.89040516025637409043425748063, 10.952340939007312349074956014208, 12.28888210220654528037378608813, 13.55251472235505711244948820672, 14.80601197051104665108633290161, 15.08068957115773076408610364556, 16.1994180684147091070690878356, 17.74953424709800648499507175903, 18.19337130824972709088030270207, 19.36192917520606892638943802702, 19.80168027425883038582077445667, 20.854122291750608599301212553048, 22.76006178177375558328524984775, 23.21836256178628840880936574288, 24.443228301802833253536883672663, 25.10892246996993324534639782814, 26.05500507364603774850664746062, 26.65896489517392865202141917951

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