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Properties

Label 1-20-20.3-r0-0-0
Degree $1$
Conductor $20$
Sign $0.850 - 0.525i$
Analytic cond. $0.0928796$
Root an. cond. $0.0928796$
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  − i·3-s + i·7-s − 9-s − 11-s + i·13-s − i·17-s + 19-s + 21-s − i·23-s + i·27-s − 29-s − 31-s + i·33-s − i·37-s + 39-s + ⋯
L(s)  = 1  − i·3-s + i·7-s − 9-s − 11-s + i·13-s − i·17-s + 19-s + 21-s − i·23-s + i·27-s − 29-s − 31-s + i·33-s − i·37-s + 39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(0.0928796\)
Root analytic conductor: \(0.0928796\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 20,\ (0:\ ),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6153549914 - 0.1748094575i\)
\(L(\frac12)\) \(\approx\) \(0.6153549914 - 0.1748094575i\)
\(L(1)\) \(\approx\) \(0.8595478356 - 0.1767421806i\)
\(L(1)\) \(\approx\) \(0.8595478356 - 0.1767421806i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 + T \)
83 \( 1 \)
89 \( 1 - iT \)
97 \( 1 \)
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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

−39.68269158589923208615540280166, −39.336264703355623494329443153673, −37.64121840189302633480402058127, −36.63925162106026851035280910428, −34.849288868370123408737536992514, −33.45096533960072786896162409934, −32.570556274132070632133243141970, −31.1632075039371300308856175534, −29.504789230302340497825221713429, −28.047119757788737551702032536619, −26.78068812058721427987306146909, −25.77843933969612115054892329945, −23.7243438948056873765175022385, −22.434540191315472933521500272742, −20.93797208528347140576261539997, −19.882153214576594928835531512624, −17.69750908991989790359894140769, −16.30388879911594151964725727174, −14.96782345013431223635605146029, −13.27477102305485924531921767602, −11.001014977110963206286226455439, −9.885690281516151356044990603413, −7.86202871799724463684935052995, −5.37092217269348812234929796648, −3.53109568390976713881226562799, 2.43970471075016573834496104089, 5.554135368747861766165490725370, 7.35262170860794421983846811052, 9.03603034832864075846403276009, 11.459596005614086402668986184279, 12.7635194118700229837484632365, 14.28943082233952163374769888088, 16.15266427433562288425121751476, 18.10099096196119455815502387985, 18.88172009903748921865918078907, 20.64653162258021723690996015926, 22.38526032224391234553804965253, 23.8957002429788890790730956947, 24.963380929599044063195818592980, 26.33239523022640549254450971302, 28.36597353127179346163501353088, 29.237289040625714209614816587000, 30.91844193420268949843068014374, 31.6515966356349751302737680839, 33.78567981564399359465343199462, 34.856507365903759726131828355361, 36.09441314324750012996248414038, 37.247801206047243723799357576921, 38.735795968667743215521103873541, 40.38883430358204640546448355975

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