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Properties

Label 1-363-363.29-r0-0-0
Degree $1$
Conductor $363$
Sign $0.628 - 0.777i$
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  + (−0.564 − 0.825i)2-s + (−0.362 + 0.931i)4-s + (0.921 − 0.389i)5-s + (0.870 + 0.491i)7-s + (0.974 − 0.226i)8-s + (−0.841 − 0.540i)10-s + (−0.774 − 0.633i)13-s + (−0.0855 − 0.996i)14-s + (−0.736 − 0.676i)16-s + (0.897 + 0.441i)17-s + (0.466 − 0.884i)19-s + (0.0285 + 0.999i)20-s + (−0.415 + 0.909i)23-s + (0.696 − 0.717i)25-s + (−0.0855 + 0.996i)26-s + ⋯
L(s)  = 1  + (−0.564 − 0.825i)2-s + (−0.362 + 0.931i)4-s + (0.921 − 0.389i)5-s + (0.870 + 0.491i)7-s + (0.974 − 0.226i)8-s + (−0.841 − 0.540i)10-s + (−0.774 − 0.633i)13-s + (−0.0855 − 0.996i)14-s + (−0.736 − 0.676i)16-s + (0.897 + 0.441i)17-s + (0.466 − 0.884i)19-s + (0.0285 + 0.999i)20-s + (−0.415 + 0.909i)23-s + (0.696 − 0.717i)25-s + (−0.0855 + 0.996i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(1.68576\)
Root analytic conductor: \(1.68576\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 363,\ (0:\ ),\ 0.628 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104768916 - 0.5272411850i\)
\(L(\frac12)\) \(\approx\) \(1.104768916 - 0.5272411850i\)
\(L(1)\) \(\approx\) \(0.9558202032 - 0.3397526293i\)
\(L(1)\) \(\approx\) \(0.9558202032 - 0.3397526293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.564 - 0.825i)T \)
5 \( 1 + (0.921 - 0.389i)T \)
7 \( 1 + (0.870 + 0.491i)T \)
13 \( 1 + (-0.774 - 0.633i)T \)
17 \( 1 + (0.897 + 0.441i)T \)
19 \( 1 + (0.466 - 0.884i)T \)
23 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.696 + 0.717i)T \)
31 \( 1 + (-0.254 + 0.967i)T \)
37 \( 1 + (-0.998 + 0.0570i)T \)
41 \( 1 + (0.941 + 0.336i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.610 - 0.791i)T \)
53 \( 1 + (0.736 - 0.676i)T \)
59 \( 1 + (-0.941 + 0.336i)T \)
61 \( 1 + (0.564 - 0.825i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.985 + 0.170i)T \)
73 \( 1 + (-0.198 - 0.980i)T \)
79 \( 1 + (-0.516 + 0.856i)T \)
83 \( 1 + (0.993 - 0.113i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.921 - 0.389i)T \)
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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

−24.60557319272578690349501980630, −24.44334688930991097575316744276, −23.144856514797953812826581544818, −22.44868494701918659392200937806, −21.23419305364713291432705387384, −20.500442909422513375650489078664, −19.215210680682693114784852954521, −18.43627089080118497108610973753, −17.64725610684323036706995346710, −16.9395321591930992485252839490, −16.18441452716525541717141020359, −14.73461843980554192339278561600, −14.311233939016977364606299610760, −13.63364442702061765372641201574, −12.10348670163407173595637002824, −10.8419509416752936767313575873, −10.02871968253565036534181318145, −9.31215352639946182454266561339, −8.01664717777938400642919292565, −7.30344791374630657098721055028, −6.22423282988369404913495430701, −5.31461132138607290601460879627, −4.28540193739788193552537405201, −2.36157035544508078724107504558, −1.22828099080581882674580842100, 1.18996857470512150847855790632, 2.14762607903536274045821118721, 3.2319250403131194374145033225, 4.826063243454166948687517101, 5.50749864347585642039310898373, 7.19988695431702624709887175091, 8.24041213364232246798567114979, 9.07389332710040962067431594577, 9.973014201462906804951515962501, 10.78076816828697306808065717741, 11.96704757479847131652589058401, 12.57066938815208171086893479394, 13.66676352416077197862632209531, 14.51005838452012633446303761700, 15.84572346011483201859328453840, 17.00419714634218027634228144686, 17.71049301458323467979780483043, 18.15970139837416605877007014629, 19.43084357523763982772021524221, 20.165279792557906121932752969052, 21.2142843248216501937548651886, 21.573324871899791367054477463557, 22.43656438861871791670877030620, 23.81306871455220909904518314618, 24.80419616108076446354120956226

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