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Properties

Label 1-363-363.74-r0-0-0
Degree $1$
Conductor $363$
Sign $-0.988 - 0.152i$
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  + (0.774 − 0.633i)2-s + (0.198 − 0.980i)4-s + (−0.897 + 0.441i)5-s + (−0.0855 − 0.996i)7-s + (−0.466 − 0.884i)8-s + (−0.415 + 0.909i)10-s + (−0.993 + 0.113i)13-s + (−0.696 − 0.717i)14-s + (−0.921 − 0.389i)16-s + (−0.564 − 0.825i)17-s + (−0.941 + 0.336i)19-s + (0.254 + 0.967i)20-s + (0.654 − 0.755i)23-s + (0.610 − 0.791i)25-s + (−0.696 + 0.717i)26-s + ⋯
L(s)  = 1  + (0.774 − 0.633i)2-s + (0.198 − 0.980i)4-s + (−0.897 + 0.441i)5-s + (−0.0855 − 0.996i)7-s + (−0.466 − 0.884i)8-s + (−0.415 + 0.909i)10-s + (−0.993 + 0.113i)13-s + (−0.696 − 0.717i)14-s + (−0.921 − 0.389i)16-s + (−0.564 − 0.825i)17-s + (−0.941 + 0.336i)19-s + (0.254 + 0.967i)20-s + (0.654 − 0.755i)23-s + (0.610 − 0.791i)25-s + (−0.696 + 0.717i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ -0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07709098062 - 1.004538615i\)
\(L(\frac12)\) \(\approx\) \(0.07709098062 - 1.004538615i\)
\(L(1)\) \(\approx\) \(0.8672552507 - 0.6544918658i\)
\(L(1)\) \(\approx\) \(0.8672552507 - 0.6544918658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.774 - 0.633i)T \)
5 \( 1 + (-0.897 + 0.441i)T \)
7 \( 1 + (-0.0855 - 0.996i)T \)
13 \( 1 + (-0.993 + 0.113i)T \)
17 \( 1 + (-0.564 - 0.825i)T \)
19 \( 1 + (-0.941 + 0.336i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.610 + 0.791i)T \)
31 \( 1 + (-0.736 - 0.676i)T \)
37 \( 1 + (-0.870 + 0.491i)T \)
41 \( 1 + (-0.998 + 0.0570i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.362 - 0.931i)T \)
53 \( 1 + (0.921 - 0.389i)T \)
59 \( 1 + (0.998 + 0.0570i)T \)
61 \( 1 + (-0.774 - 0.633i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.0285 + 0.999i)T \)
73 \( 1 + (-0.974 + 0.226i)T \)
79 \( 1 + (0.985 + 0.170i)T \)
83 \( 1 + (0.516 - 0.856i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.897 + 0.441i)T \)
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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

−24.9055697463804457504589670433, −24.210338182035349376742811811657, −23.48705894623526155195603688916, −22.58316173661774338007669038561, −21.718301725148273893632662686452, −21.07102305336216097997937067550, −19.769838399060987295421226375619, −19.18878264663822018247906710470, −17.74668089282121325358621135346, −16.98631859411694273253316532575, −15.955396523257679668008860783332, −15.218015592156661279041522697346, −14.7561769776989795043061433229, −13.31104784957778352466529436567, −12.494695333926505196650004693289, −11.938117818188668382979937978, −10.87998745541101410669586453242, −9.13717999017325192556358830966, −8.42646474945260757704789939704, −7.46246366042955208663258642180, −6.43098407168563434101186450673, −5.283618434885959906577902606579, −4.50688287859437857133391932986, −3.37741637932375086875024707512, −2.20430816429574161887819368567, 0.44566140668807713510211168947, 2.20466449661163711967607854996, 3.3398598530681898941413669443, 4.247561288473945327314891104382, 5.05850485386443371955490264500, 6.748423765325361076990295249833, 7.164007679932187856917760190711, 8.68028441085013415033723801433, 10.09247977725293132469608500825, 10.70820814663501661557954984162, 11.63351464339065087405541067027, 12.46654955677612357246167031892, 13.44674843341584693730548987910, 14.42573444318962579962046187335, 15.04303814783475958885629381316, 16.09991644371732288653316131640, 17.083628696890652459179585393136, 18.48279001390587671578102143489, 19.21699754859795578117622578920, 20.05555688600786928326116746995, 20.574779341201711134725801514181, 21.85849569820079434948216761483, 22.55497764739494005740569574197, 23.323611064720010878672695953070, 23.93266923018949828847112354606

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