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Properties

Label 1-363-363.23-r1-0-0
Degree $1$
Conductor $363$
Sign $0.987 - 0.155i$
Analytic cond. $39.0097$
Root an. cond. $39.0097$
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  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)13-s + (−0.841 − 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.654 − 0.755i)20-s + (0.959 + 0.281i)23-s + (0.415 + 0.909i)25-s + (−0.841 + 0.540i)26-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.841 − 0.540i)5-s + (−0.959 + 0.281i)7-s + (−0.841 + 0.540i)8-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)13-s + (−0.841 − 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.415 − 0.909i)17-s + (0.415 − 0.909i)19-s + (0.654 − 0.755i)20-s + (0.959 + 0.281i)23-s + (0.415 + 0.909i)25-s + (−0.841 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(39.0097\)
Root analytic conductor: \(39.0097\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (1:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.273316267 - 0.09938059058i\)
\(L(\frac12)\) \(\approx\) \(1.273316267 - 0.09938059058i\)
\(L(1)\) \(\approx\) \(0.9701482029 + 0.3522092420i\)
\(L(1)\) \(\approx\) \(0.9701482029 + 0.3522092420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.654 + 0.755i)T \)
5 \( 1 + (-0.841 - 0.540i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.415 - 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.654 - 0.755i)T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (0.654 - 0.755i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.415 + 0.909i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
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

−24.24958445098247315541663865745, −23.27011670245382397065192706878, −22.67944937737073574524206640336, −22.18181950360243149185934529908, −20.92000179958825449146781431565, −20.07034956287612856697223229874, −19.31984812050734382836474134189, −18.79010441487018845990355811667, −17.5879730147325323955189837618, −16.21711085400844880211732608852, −15.34781006507974186200791217097, −14.68499882963097067767424285339, −13.52880419516992407281222515336, −12.67047238141378197052764112014, −11.9862274715139188764037422961, −10.72251962550003050632517402231, −10.34677121436910487245454342762, −9.09586939081381571312577020891, −7.74118418944167171787918868774, −6.61698637221885270731622162247, −5.684220213178051210924582547994, −4.27958123403006972639464905389, −3.45335821929938283640141802542, −2.65575518035491609998814604778, −0.914639398065330568960872572117, 0.3689746582379948403911822845, 2.61066337326994195664297726274, 3.69746117351347321147216008020, 4.63968405333362955301939792552, 5.60670469484025640349687513984, 6.93414866809131349553929242203, 7.38976321998721986184557073251, 8.94315930142847966606606840111, 9.228718042004896497578296228820, 11.2253112557040567216763605470, 11.95897032702007793389719063879, 12.89404919807855881307518462054, 13.56993257968901572215229560863, 14.780736493499504051970375316890, 15.6610352665933789646846111137, 16.25331450834388835307937777793, 16.94562715803985554306690355023, 18.21144001716500216734197985094, 19.23540202133119757132441594781, 20.12179276324712808292566571640, 21.11844163445597563176500636597, 22.11884240879137616722603892490, 22.79983753849500697286075013210, 23.64596455935379495519534496804, 24.36910343873116045154881581650

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