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Properties

Label 1-4020-4020.1067-r1-0-0
Degree $1$
Conductor $4020$
Sign $0.887 + 0.460i$
Analytic cond. $432.008$
Root an. cond. $432.008$
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  + (−0.989 + 0.142i)7-s + (0.841 − 0.540i)11-s + (−0.909 − 0.415i)13-s + (−0.281 + 0.959i)17-s + (−0.142 + 0.989i)19-s + (0.755 + 0.654i)23-s + 29-s + (−0.415 − 0.909i)31-s + i·37-s + (0.959 + 0.281i)41-s + (0.281 − 0.959i)43-s + (−0.755 − 0.654i)47-s + (0.959 − 0.281i)49-s + (−0.281 − 0.959i)53-s + (−0.415 − 0.909i)59-s + ⋯
L(s)  = 1  + (−0.989 + 0.142i)7-s + (0.841 − 0.540i)11-s + (−0.909 − 0.415i)13-s + (−0.281 + 0.959i)17-s + (−0.142 + 0.989i)19-s + (0.755 + 0.654i)23-s + 29-s + (−0.415 − 0.909i)31-s + i·37-s + (0.959 + 0.281i)41-s + (0.281 − 0.959i)43-s + (−0.755 − 0.654i)47-s + (0.959 − 0.281i)49-s + (−0.281 − 0.959i)53-s + (−0.415 − 0.909i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(432.008\)
Root analytic conductor: \(432.008\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1067, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4020,\ (1:\ ),\ 0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.504360988 + 0.3671251269i\)
\(L(\frac12)\) \(\approx\) \(1.504360988 + 0.3671251269i\)
\(L(1)\) \(\approx\) \(0.9276629232 + 0.02862587984i\)
\(L(1)\) \(\approx\) \(0.9276629232 + 0.02862587984i\)

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 \( 1 + (-0.989 + 0.142i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.909 - 0.415i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.959 + 0.281i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.755 - 0.654i)T \)
53 \( 1 + (-0.281 - 0.959i)T \)
59 \( 1 + (-0.415 - 0.909i)T \)
61 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.540 - 0.841i)T \)
79 \( 1 + (0.415 - 0.909i)T \)
83 \( 1 + (-0.540 - 0.841i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + iT \)
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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

−18.140849446632891420984080451806, −17.56674196984519635029999584774, −16.88191298014438358397373810302, −16.193100488531555519469843620938, −15.67597622856950629421617976661, −14.754693346984151703912952884648, −14.21746608159888966895819807599, −13.50738690491704137529976603357, −12.49994287167946848844219117947, −12.400371009527438755554794872193, −11.33024182559998356893099340961, −10.72301178184363005636097096547, −9.69684807385704762139727420688, −9.36437820137127397114415115454, −8.75234479464285171370427182359, −7.5494899404846307692369311395, −6.80208613553037209390474917560, −6.66202372917492341965416543160, −5.46938990594435863218845175958, −4.59554615785867229198149443533, −4.12416767130947398783615192696, −2.86119677173847710282591531159, −2.59249480229851108810193840513, −1.294156427637637793639338757638, −0.40347566677205935596255244959, 0.520259888637107890567658471482, 1.52036812081132213744193401896, 2.48518328855027599283376138501, 3.367672647515549954647974383362, 3.87083488725471336001578750186, 4.88126412433602981355856157978, 5.80826231979480082876594126980, 6.327810490226427642499983733042, 7.03889381982623861328496526968, 7.93160351081531883637775364592, 8.652135862852916122664625825153, 9.417176634876394355035451212662, 10.00925091710596544981648693005, 10.66458581312411008074935342570, 11.62632741241624952965878576028, 12.18455020944983204578444426767, 12.94408087196449010652862670413, 13.409555496478935701810496600348, 14.43925960013004875510766714018, 14.873554158197690923507307102249, 15.6675471328736877447227484879, 16.381596531711331975343006645979, 17.04886476006381703209205011382, 17.446358814996112200663944536200, 18.50056951264160154011922127452

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