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Properties

Label 1-20-20.7-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: $\chi_{20} (7, \cdot )$
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

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

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