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Properties

Label 1-4140-4140.2363-r0-0-0
Degree $1$
Conductor $4140$
Sign $-0.646 + 0.763i$
Analytic cond. $19.2260$
Root an. cond. $19.2260$
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  + (−0.690 − 0.723i)7-s + (0.327 + 0.945i)11-s + (−0.690 + 0.723i)13-s + (−0.989 + 0.142i)17-s + (0.142 − 0.989i)19-s + (0.928 + 0.371i)29-s + (0.888 + 0.458i)31-s + (−0.909 + 0.415i)37-s + (0.995 − 0.0950i)41-s + (−0.458 − 0.888i)43-s + (0.866 + 0.5i)47-s + (−0.0475 + 0.998i)49-s + (0.281 − 0.959i)53-s + (−0.723 − 0.690i)59-s + (−0.0475 − 0.998i)61-s + ⋯
L(s)  = 1  + (−0.690 − 0.723i)7-s + (0.327 + 0.945i)11-s + (−0.690 + 0.723i)13-s + (−0.989 + 0.142i)17-s + (0.142 − 0.989i)19-s + (0.928 + 0.371i)29-s + (0.888 + 0.458i)31-s + (−0.909 + 0.415i)37-s + (0.995 − 0.0950i)41-s + (−0.458 − 0.888i)43-s + (0.866 + 0.5i)47-s + (−0.0475 + 0.998i)49-s + (0.281 − 0.959i)53-s + (−0.723 − 0.690i)59-s + (−0.0475 − 0.998i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.646 + 0.763i$
Analytic conductor: \(19.2260\)
Root analytic conductor: \(19.2260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2363, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4140,\ (0:\ ),\ -0.646 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2350400869 + 0.5069150053i\)
\(L(\frac12)\) \(\approx\) \(0.2350400869 + 0.5069150053i\)
\(L(1)\) \(\approx\) \(0.8332835151 + 0.04844455950i\)
\(L(1)\) \(\approx\) \(0.8332835151 + 0.04844455950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.690 - 0.723i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (-0.690 + 0.723i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (0.142 - 0.989i)T \)
29 \( 1 + (0.928 + 0.371i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (-0.909 + 0.415i)T \)
41 \( 1 + (0.995 - 0.0950i)T \)
43 \( 1 + (-0.458 - 0.888i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 + (0.281 - 0.959i)T \)
59 \( 1 + (-0.723 - 0.690i)T \)
61 \( 1 + (-0.0475 - 0.998i)T \)
67 \( 1 + (0.945 + 0.327i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.989 - 0.142i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.0950 + 0.995i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (-0.814 + 0.580i)T \)
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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

−18.20002989069760514313983127769, −17.5037366439443374184563873664, −16.79549396105998853033873197588, −16.04752350873557650601943874159, −15.555611237263544090990327129996, −14.85890508248497223649110381768, −14.040544696536594028007810893107, −13.41837311909296618848942142998, −12.65580571175787352005769194414, −12.0404257662791759582407413942, −11.451917443465077760223644919801, −10.465941808092840250456151322441, −9.96688344055040615713309633637, −9.06904565909741494149215949787, −8.57509986962026216094010254262, −7.77512751905721954018143716030, −6.91073143105740235506006755703, −6.02463745037994291024118255456, −5.74366777473905026004614229512, −4.67905388987067211332882049241, −3.87035609693259018597360338945, −2.89931385111560091867267647941, −2.52580953463189669475417587592, −1.28754665816972569506115900462, −0.1709821446681488021607904389, 1.039532816044826260273295620363, 2.079994138089146745584748106510, 2.791934923516007425388155397850, 3.7869753340891200862961672289, 4.53281263054073430068061193113, 4.983007327481567676289950379766, 6.29197284888786101420958736661, 6.93021272736255543756420470851, 7.15287405787788482407434390762, 8.302790295714556525450690754543, 9.130462193463541061874501072662, 9.66349930335096628108550464990, 10.365492479330175792876370849813, 11.05014655218168294083808520951, 11.949879712996307999361500467190, 12.48262140127173271953708753582, 13.24832954566731583659087723952, 13.92340639669774738752135589015, 14.46095738742948765197685548864, 15.53676774719508330496363999766, 15.74734967744582964021201826839, 16.757029360314048300464028548554, 17.40704002173323958200898562730, 17.68544345982413416903954523503, 18.78100645259000892097008460493

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