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Properties

Label 1-363-363.2-r0-0-0
Degree $1$
Conductor $363$
Sign $0.889 + 0.457i$
Analytic cond. $1.68576$
Root an. cond. $1.68576$
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  + (−0.998 − 0.0570i)2-s + (0.993 + 0.113i)4-s + (0.466 + 0.884i)5-s + (0.921 + 0.389i)7-s + (−0.985 − 0.170i)8-s + (−0.415 − 0.909i)10-s + (0.870 − 0.491i)13-s + (−0.897 − 0.441i)14-s + (0.974 + 0.226i)16-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.362 + 0.931i)20-s + (0.654 + 0.755i)23-s + (−0.564 + 0.825i)25-s + (−0.897 + 0.441i)26-s + ⋯
L(s)  = 1  + (−0.998 − 0.0570i)2-s + (0.993 + 0.113i)4-s + (0.466 + 0.884i)5-s + (0.921 + 0.389i)7-s + (−0.985 − 0.170i)8-s + (−0.415 − 0.909i)10-s + (0.870 − 0.491i)13-s + (−0.897 − 0.441i)14-s + (0.974 + 0.226i)16-s + (0.941 − 0.336i)17-s + (0.0285 − 0.999i)19-s + (0.362 + 0.931i)20-s + (0.654 + 0.755i)23-s + (−0.564 + 0.825i)25-s + (−0.897 + 0.441i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ 0.889 + 0.457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046814727 + 0.2534749115i\)
\(L(\frac12)\) \(\approx\) \(1.046814727 + 0.2534749115i\)
\(L(1)\) \(\approx\) \(0.8899517107 + 0.1195639771i\)
\(L(1)\) \(\approx\) \(0.8899517107 + 0.1195639771i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.998 - 0.0570i)T \)
5 \( 1 + (0.466 + 0.884i)T \)
7 \( 1 + (0.921 + 0.389i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (0.941 - 0.336i)T \)
19 \( 1 + (0.0285 - 0.999i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.564 - 0.825i)T \)
31 \( 1 + (0.198 - 0.980i)T \)
37 \( 1 + (-0.736 + 0.676i)T \)
41 \( 1 + (-0.254 - 0.967i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.774 + 0.633i)T \)
53 \( 1 + (-0.974 + 0.226i)T \)
59 \( 1 + (0.254 - 0.967i)T \)
61 \( 1 + (0.998 - 0.0570i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.610 + 0.791i)T \)
73 \( 1 + (-0.516 + 0.856i)T \)
79 \( 1 + (-0.696 + 0.717i)T \)
83 \( 1 + (0.0855 - 0.996i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.466 + 0.884i)T \)
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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

−24.849632485875522621958920019407, −23.94350520740442811638701171068, −23.275319741931922079144679439682, −21.52729861522742395870071717041, −20.83388894493212937395887967301, −20.44350327510550135472669145778, −19.22883778534132075142334264822, −18.33142912502026778402942883715, −17.55538510460848266661937005171, −16.636100480222602634402247662080, −16.24121973560228776158450286681, −14.82037405204525977388308344612, −14.036555278695540223854881339977, −12.71374183191425023473166869356, −11.82009355488353174757248345487, −10.76753961511028773823343417762, −9.998104017839168159380485922, −8.77715727013792742178258667046, −8.332048509052769480603992393575, −7.199371580702637328150036953106, −5.993297084961581579726246599680, −5.018558776802310831841553121472, −3.573125814861678234483820739934, −1.81326866843440911549352559946, −1.18170107342350996755811969026, 1.27330193890554719441661246906, 2.4181098607966170531362534675, 3.40999567623030047200832453611, 5.31199773591125821560199866658, 6.23768529576992122435553806364, 7.35701293791916256849919124121, 8.13810873002064111125499194884, 9.24401845003836589789046811852, 10.09893870697112189855376948974, 11.208491067089813849673923565215, 11.4986002176448872985530127961, 13.03064926999733300350013456525, 14.20536870435088248190369465907, 15.14791334528177638804790558414, 15.78559819147150700243780788507, 17.266247988714793084898290246671, 17.62729522496942881301402045076, 18.65815210325997067975989552438, 19.05107262669978449469802917529, 20.50242894854740550444808692717, 21.04502144632528534493121122525, 21.92107712046452568127913830437, 23.05721812167904904695297358604, 24.145529089157463949739781357653, 25.03698326536453198871535663578

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