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Properties

Label 1-363-363.200-r0-0-0
Degree $1$
Conductor $363$
Sign $-0.464 - 0.885i$
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  + (0.696 − 0.717i)2-s + (−0.0285 − 0.999i)4-s + (0.870 + 0.491i)5-s + (−0.774 − 0.633i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (−0.610 − 0.791i)13-s + (−0.993 + 0.113i)14-s + (−0.998 + 0.0570i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (0.466 − 0.884i)20-s + (−0.841 − 0.540i)23-s + (0.516 + 0.856i)25-s + (−0.993 − 0.113i)26-s + ⋯
L(s)  = 1  + (0.696 − 0.717i)2-s + (−0.0285 − 0.999i)4-s + (0.870 + 0.491i)5-s + (−0.774 − 0.633i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (−0.610 − 0.791i)13-s + (−0.993 + 0.113i)14-s + (−0.998 + 0.0570i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (0.466 − 0.884i)20-s + (−0.841 − 0.540i)23-s + (0.516 + 0.856i)25-s + (−0.993 − 0.113i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.464 - 0.885i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (200, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ -0.464 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9264999911 - 1.531927030i\)
\(L(\frac12)\) \(\approx\) \(0.9264999911 - 1.531927030i\)
\(L(1)\) \(\approx\) \(1.225978770 - 0.8428959433i\)
\(L(1)\) \(\approx\) \(1.225978770 - 0.8428959433i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.696 - 0.717i)T \)
5 \( 1 + (0.870 + 0.491i)T \)
7 \( 1 + (-0.774 - 0.633i)T \)
13 \( 1 + (-0.610 - 0.791i)T \)
17 \( 1 + (0.0855 - 0.996i)T \)
19 \( 1 + (0.921 - 0.389i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (0.516 - 0.856i)T \)
31 \( 1 + (0.941 + 0.336i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (0.897 + 0.441i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.985 + 0.170i)T \)
53 \( 1 + (0.998 + 0.0570i)T \)
59 \( 1 + (-0.897 + 0.441i)T \)
61 \( 1 + (-0.696 - 0.717i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.974 - 0.226i)T \)
73 \( 1 + (0.254 - 0.967i)T \)
79 \( 1 + (-0.198 + 0.980i)T \)
83 \( 1 + (-0.362 + 0.931i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (-0.870 + 0.491i)T \)
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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

−24.80603480008761882312917576455, −24.30526839446293150930840934866, −23.333273515279909639112252508364, −22.12457387829480800402885426152, −21.83801488110530618246177074613, −20.927140687735813536452995149785, −19.85159289285704626766020727797, −18.66100849860920780749202194305, −17.63280448395026542655082255344, −16.83126155972820572891602777952, −16.09657449274585379780969583505, −15.20034708623811368484314882353, −14.088758316072416245638421048270, −13.501894896819057300870617793984, −12.3805201794025882021874724004, −11.98206598617152330315448180086, −10.18187579364154489940636792372, −9.253457872093663288675404468954, −8.41518920283294678454015846564, −7.110180949963672017772690512819, −6.10810405744912626701883136493, −5.492625477376492761699554459310, −4.348440479331554413282166718387, −3.09693968669338882416321106856, −1.918795267130733682909780268242, 0.873230718195968506044837038059, 2.512604201643692195163720480087, 3.10238075287005465972920117893, 4.46228970793210777228235463128, 5.566110375233393498277394102295, 6.45104750901219531735645453369, 7.44984763361878486821908364897, 9.32006777912583152426162052951, 9.98963469499176146868896315556, 10.61852463885807537136592886936, 11.82465963036368825508786252433, 12.76457739851051644870841750881, 13.72899235834000504194950168846, 14.10372390406021601138383635875, 15.31787347493788813968772681088, 16.25602813785697370415271816423, 17.580888909146558659024128458666, 18.28649212256198234440039003343, 19.38966116935206081772168931752, 20.10523400855309144068669532657, 20.94386652556000571371083026968, 21.90870641181213743262284272454, 22.66462371655049568753701008404, 23.04948023281624834816992086212, 24.508981159721313283753569021130

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