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Properties

Label 1-483-483.26-r0-0-0
Degree $1$
Conductor $483$
Sign $-0.876 - 0.481i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
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  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.786 + 0.618i)5-s + (0.654 − 0.755i)8-s + (0.995 + 0.0950i)10-s + (−0.723 + 0.690i)11-s + (−0.415 − 0.909i)13-s + (−0.995 + 0.0950i)16-s + (−0.888 + 0.458i)17-s + (0.888 + 0.458i)19-s + (−0.654 − 0.755i)20-s + 22-s + (0.235 − 0.971i)25-s + (−0.327 + 0.945i)26-s + (−0.841 − 0.540i)29-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.0475 + 0.998i)4-s + (−0.786 + 0.618i)5-s + (0.654 − 0.755i)8-s + (0.995 + 0.0950i)10-s + (−0.723 + 0.690i)11-s + (−0.415 − 0.909i)13-s + (−0.995 + 0.0950i)16-s + (−0.888 + 0.458i)17-s + (0.888 + 0.458i)19-s + (−0.654 − 0.755i)20-s + 22-s + (0.235 − 0.971i)25-s + (−0.327 + 0.945i)26-s + (−0.841 − 0.540i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ -0.876 - 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05570880467 - 0.2168738478i\)
\(L(\frac12)\) \(\approx\) \(0.05570880467 - 0.2168738478i\)
\(L(1)\) \(\approx\) \(0.4935310914 - 0.1086801151i\)
\(L(1)\) \(\approx\) \(0.4935310914 - 0.1086801151i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
5 \( 1 + (-0.786 + 0.618i)T \)
11 \( 1 + (-0.723 + 0.690i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.888 + 0.458i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.841 - 0.540i)T \)
31 \( 1 + (0.327 + 0.945i)T \)
37 \( 1 + (0.928 - 0.371i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.580 - 0.814i)T \)
59 \( 1 + (-0.995 - 0.0950i)T \)
61 \( 1 + (-0.981 - 0.189i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (-0.0475 - 0.998i)T \)
79 \( 1 + (0.580 - 0.814i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.327 + 0.945i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−24.271911423221711488550570478703, −23.6319115905968425143751650366, −22.6911193685747249909786688937, −21.57714588230203443187056009868, −20.34884490377991572016341343256, −19.84812506158043667405363495985, −18.83239675083548888707895577345, −18.27439225449460668489146280190, −17.07948700015968232803003906870, −16.37452153987477513801717531879, −15.7670212868479783083691746737, −14.94534814648215176446911126124, −13.83553521340637352930904904060, −12.99581055358171243385209014893, −11.51374925155476844059126619702, −11.14913703064886699630323982168, −9.67491527638051615973826312313, −9.04147103462262524114102196222, −8.05004782916623456119511081578, −7.38935237146444489441417862560, −6.314715805227093602667694357716, −5.13012638950349891136020507702, −4.39859015243336879125270448949, −2.763304271446707433798218670545, −1.24110426918396120032400952026, 0.16744815629039619306488589387, 1.9477942266918497091510212107, 2.96729246611044405280470344113, 3.85833199952002145177259865420, 5.05464754041985870299547612455, 6.68626381884564630614231607989, 7.63616061288272343557761967414, 8.13782701429912267511291253639, 9.4249904177464377765784849355, 10.365520982184086258123342568405, 10.93952721849540907100397618731, 11.992190038666509000861454848842, 12.64858887215680525562525806848, 13.68380267809648133767435213041, 15.096603186812992216146743481870, 15.619325293709000260141383813286, 16.690662776335500963246543363198, 17.81543338597995433921391541026, 18.2356658626065164876291314109, 19.21823659977869423354656528654, 20.00750019487735087950456228439, 20.53478040386447339558805577095, 21.70407607011667954406540852382, 22.51141812895764574146962550413, 23.12434354620742084439329315115

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