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Properties

Label 1-4020-4020.2519-r0-0-0
Degree $1$
Conductor $4020$
Sign $-0.997 - 0.0747i$
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  + (−0.142 + 0.989i)7-s + (0.841 + 0.540i)11-s + (−0.415 − 0.909i)13-s + (−0.959 + 0.281i)17-s + (0.142 + 0.989i)19-s + (0.654 + 0.755i)23-s − 29-s + (−0.415 + 0.909i)31-s − 37-s + (0.959 − 0.281i)41-s + (−0.959 + 0.281i)43-s + (0.654 + 0.755i)47-s + (−0.959 − 0.281i)49-s + (−0.959 − 0.281i)53-s + (0.415 − 0.909i)59-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)7-s + (0.841 + 0.540i)11-s + (−0.415 − 0.909i)13-s + (−0.959 + 0.281i)17-s + (0.142 + 0.989i)19-s + (0.654 + 0.755i)23-s − 29-s + (−0.415 + 0.909i)31-s − 37-s + (0.959 − 0.281i)41-s + (−0.959 + 0.281i)43-s + (0.654 + 0.755i)47-s + (−0.959 − 0.281i)49-s + (−0.959 − 0.281i)53-s + (0.415 − 0.909i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.997 - 0.0747i$
Analytic conductor: \(18.6688\)
Root analytic conductor: \(18.6688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (2519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (0:\ ),\ -0.997 - 0.0747i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01924574896 + 0.5140056104i\)
\(L(\frac12)\) \(\approx\) \(0.01924574896 + 0.5140056104i\)
\(L(1)\) \(\approx\) \(0.8563691729 + 0.2124819747i\)
\(L(1)\) \(\approx\) \(0.8563691729 + 0.2124819747i\)

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 \( 1 + (-0.142 + 0.989i)T \)
11 \( 1 + (0.841 + 0.540i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 - T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (0.654 + 0.755i)T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (-0.841 + 0.540i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.841 - 0.540i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
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

−18.04217079477364341014356644190, −17.32026895845745117319688043213, −16.73874077470769191637421118743, −16.33790614064344800680556498048, −15.358963460578145750033922865, −14.66337688238029637707620066884, −13.97430899930191622628434111150, −13.41642577655861866922958277934, −12.81452079525842727887291084998, −11.73626143259917104319913392874, −11.27481502247243958953037908116, −10.66992625711536480484090753461, −9.71603617328401686448047226447, −9.09418459262025221379545382700, −8.56089588400985586449675018633, −7.329225415799833491503059617198, −6.98656717558660249557592951450, −6.3379881253136830334151495205, −5.309649989873606401662304986960, −4.367308188293396711093088774630, −4.01262649522165493958706321651, −2.99901272626592054638659038686, −2.10864033469792422166798103661, −1.14232756560577400334988956736, −0.144067824290725378987955016296, 1.44661469844599012183994384754, 2.036500538899085270133171864247, 3.06302068650947318610092665179, 3.69674075167416146873896272893, 4.71611770160843742012103773776, 5.43703664076612651951505998672, 6.0752339356825362790751937100, 6.916872920936571288205442689190, 7.618191768645414937440861610764, 8.50846842089029405314541747113, 9.12821530167399664301589965648, 9.72409480256844632900646401553, 10.53834859873676522724509122824, 11.36811095898068763615796969255, 11.99379457827783799988603045417, 12.74375436217698534937616137081, 13.075057400978182146610287212263, 14.34937847403543573480750412934, 14.66307044744755631526895490642, 15.50850053004682804368292324161, 15.88852412176215781876912107836, 16.91677151274582528280282808371, 17.56489293476646289069002959994, 17.97363815585748800704886869683, 18.99951529231006948618245783230

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