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Properties

Label 1-483-483.149-r0-0-0
Degree $1$
Conductor $483$
Sign $0.0337 - 0.999i$
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  + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.888 − 0.458i)5-s + (0.959 − 0.281i)8-s + (−0.928 − 0.371i)10-s + (−0.995 − 0.0950i)11-s + (−0.142 − 0.989i)13-s + (0.928 − 0.371i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (−0.959 − 0.281i)20-s − 22-s + (0.580 + 0.814i)25-s + (−0.235 − 0.971i)26-s + (0.654 − 0.755i)29-s + ⋯
L(s)  = 1  + (0.995 − 0.0950i)2-s + (0.981 − 0.189i)4-s + (−0.888 − 0.458i)5-s + (0.959 − 0.281i)8-s + (−0.928 − 0.371i)10-s + (−0.995 − 0.0950i)11-s + (−0.142 − 0.989i)13-s + (0.928 − 0.371i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (−0.959 − 0.281i)20-s − 22-s + (0.580 + 0.814i)25-s + (−0.235 − 0.971i)26-s + (0.654 − 0.755i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.0337 - 0.999i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ 0.0337 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.380260557 - 1.334486904i\)
\(L(\frac12)\) \(\approx\) \(1.380260557 - 1.334486904i\)
\(L(1)\) \(\approx\) \(1.477342518 - 0.5243076684i\)
\(L(1)\) \(\approx\) \(1.477342518 - 0.5243076684i\)

Euler product

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

−23.62454585917787480193553433579, −23.37001517979451946498323418353, −22.338609046002541419246804174488, −21.55197334747255649599548190327, −20.773631031899933842684697437326, −19.78339839975695716247709820590, −19.1048239901839182633269751840, −18.1027812994563324539510155947, −16.78644257251259840137686237584, −15.983110253006125899460360642197, −15.34188104929244006630324507825, −14.442112078201576322855485371197, −13.74011376926148463296488305208, −12.48075418789343210395790356429, −12.06819537022238396926385927364, −10.87733209828190091172093012921, −10.38810109982473961019588114691, −8.65664590225877018214756289059, −7.66004654107723509564813038208, −6.93748406269688395764405992096, −5.90509636005681134586806195356, −4.761514429437286054173002034498, −3.90302106027850712613201048834, −2.97924965378603160348801221931, −1.80638558778212836944718459146, 0.746366136793205666928937292810, 2.546577694960139297122910494308, 3.280548395074816343066903582202, 4.6326001411972295810414771730, 5.07657373355706975306063647535, 6.32148621221549144225941512951, 7.53978596327963612277037467675, 8.05755953472396168390947764486, 9.549257601228716847013229106989, 10.72934590163769478968241913254, 11.4626901783273896157517251508, 12.32358131673565916434587699339, 13.15156172667828215969282579423, 13.81002731418200502530776551786, 15.25425156058440698487254175076, 15.50721866559201009782773767471, 16.33269303102715805113353348063, 17.46143436721774754527672869726, 18.65996865582618039817905113681, 19.60107857185853181767249138758, 20.40504024455080198383000816114, 20.847768281582534975659653968709, 22.069588020177026513150571848629, 22.75695513770926965791657379877, 23.52054351676547422110889154900

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