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Properties

Label 1-4020-4020.1019-r0-0-0
Degree $1$
Conductor $4020$
Sign $0.463 - 0.886i$
Analytic cond. $18.6688$
Root an. cond. $18.6688$
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  + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)11-s + (−0.841 + 0.540i)13-s + (−0.142 + 0.989i)17-s + (0.654 + 0.755i)19-s + (−0.415 − 0.909i)23-s − 29-s + (−0.841 − 0.540i)31-s − 37-s + (0.142 − 0.989i)41-s + (−0.142 + 0.989i)43-s + (−0.415 − 0.909i)47-s + (−0.142 − 0.989i)49-s + (−0.142 − 0.989i)53-s + (0.841 + 0.540i)59-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)7-s + (−0.959 + 0.281i)11-s + (−0.841 + 0.540i)13-s + (−0.142 + 0.989i)17-s + (0.654 + 0.755i)19-s + (−0.415 − 0.909i)23-s − 29-s + (−0.841 − 0.540i)31-s − 37-s + (0.142 − 0.989i)41-s + (−0.142 + 0.989i)43-s + (−0.415 − 0.909i)47-s + (−0.142 − 0.989i)49-s + (−0.142 − 0.989i)53-s + (0.841 + 0.540i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.463 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.463 - 0.886i$
Analytic conductor: \(18.6688\)
Root analytic conductor: \(18.6688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1019, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (0:\ ),\ 0.463 - 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4074666060 - 0.2466893534i\)
\(L(\frac12)\) \(\approx\) \(0.4074666060 - 0.2466893534i\)
\(L(1)\) \(\approx\) \(0.7306766362 + 0.1079431710i\)
\(L(1)\) \(\approx\) \(0.7306766362 + 0.1079431710i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
67 \( 1 \)
good7 \( 1 + (-0.654 + 0.755i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.142 + 0.989i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 - T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (-0.415 - 0.909i)T \)
53 \( 1 + (-0.142 - 0.989i)T \)
59 \( 1 + (0.841 + 0.540i)T \)
61 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.959 + 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.959 - 0.281i)T \)
89 \( 1 + (-0.415 + 0.909i)T \)
97 \( 1 - T \)
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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

−18.51210349779714289963324393111, −17.93148316449589263260471770410, −17.23284038393908009742530251947, −16.52954496964819139251430271341, −15.79281538286085378068043404048, −15.45753937168375728690865183167, −14.373130835647347682615520909156, −13.74816555824650933791720605069, −13.16022485816521636562772270135, −12.58845184174797900846810083888, −11.66903639168666650822322011807, −10.97682127298010624093144898198, −10.27458934818690249931120618201, −9.62401790007768619422660742484, −9.058130865310469329243867524729, −7.9046842949878538039254654034, −7.36908483717838558830616943130, −6.90044515814681296958312969405, −5.72247303349225914387746859254, −5.20912440025588301277237442630, −4.40771803760487297682552246255, −3.28469543978359441624860487329, −2.9673283521220428738028583855, −1.88266721975901313566951234984, −0.68086655602526051381071034952, 0.18160405926338829500389661845, 1.871972297304432028580483355769, 2.2396224207617397480384076372, 3.27810677842689332655076319366, 3.97448340311574751236929997794, 5.02398897567699416944321968070, 5.58910923155122787825194136914, 6.34705514417180970147310325577, 7.15450053084432233914239130589, 7.89404798146778700905528685044, 8.62491388070428693490074850673, 9.42894901986705027280270267954, 10.01552165520866581822415946402, 10.65629839331063922074552929148, 11.598532235255119871764294338788, 12.35606298472706653282854089363, 12.7282176576348096938037818210, 13.471089235435774106180582836087, 14.430467817170578538427980719971, 14.96805387126925838319667043223, 15.60891281631140942134503038726, 16.417834139000850619868123913750, 16.79169685386493811480729833956, 17.83460558446555444857810552961, 18.397386105999571813167948438125

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