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Properties

Label 1-1980-1980.203-r1-0-0
Degree $1$
Conductor $1980$
Sign $-0.739 + 0.673i$
Analytic cond. $212.780$
Root an. cond. $212.780$
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  + (−0.743 + 0.669i)7-s + (−0.406 + 0.913i)13-s + (0.587 − 0.809i)17-s + (0.309 + 0.951i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)29-s + (−0.913 − 0.406i)31-s + (−0.951 − 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.104 − 0.994i)49-s + (0.587 + 0.809i)53-s + (0.978 − 0.207i)59-s + (0.913 − 0.406i)61-s + ⋯
L(s)  = 1  + (−0.743 + 0.669i)7-s + (−0.406 + 0.913i)13-s + (0.587 − 0.809i)17-s + (0.309 + 0.951i)19-s + (0.866 − 0.5i)23-s + (0.669 + 0.743i)29-s + (−0.913 − 0.406i)31-s + (−0.951 − 0.309i)37-s + (−0.669 + 0.743i)41-s + (0.866 + 0.5i)43-s + (−0.207 + 0.978i)47-s + (0.104 − 0.994i)49-s + (0.587 + 0.809i)53-s + (0.978 − 0.207i)59-s + (0.913 − 0.406i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.739 + 0.673i$
Analytic conductor: \(212.780\)
Root analytic conductor: \(212.780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1980,\ (1:\ ),\ -0.739 + 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4570550721 + 1.180211974i\)
\(L(\frac12)\) \(\approx\) \(0.4570550721 + 1.180211974i\)
\(L(1)\) \(\approx\) \(0.9257803540 + 0.2119679845i\)
\(L(1)\) \(\approx\) \(0.9257803540 + 0.2119679845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good7 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.406 + 0.913i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.951 + 0.309i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (0.406 + 0.913i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.994 + 0.104i)T \)
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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

−19.41189187792935843078207746699, −19.08885190369612713540838981758, −17.87432458239327766150958937082, −17.33075122414330668529333916998, −16.7051379707233539575536483098, −15.786278494424023864937416518907, −15.24223376575158032761841224603, −14.375541533825878048274543980542, −13.49120518913092599955263375735, −12.970651627896697197573285513242, −12.24022443867687891490353932288, −11.31393494895741506973200886195, −10.37541253413372485934508261074, −10.03150216647515364941650657091, −9.03160134246299166175126894634, −8.23821884777451433541978519784, −7.22714988686037302430711013402, −6.83922537916665515665305552702, −5.66499968852807534771687887495, −5.07670374223884443301366075945, −3.86582894265302468629903376404, −3.30044007364079667711924340202, −2.33961101754893205686598951157, −1.052274579031774098888319660864, −0.27195876226714400504858951119, 0.983576393663732115364512515600, 2.10848192167064474532002661835, 2.96349433903203716771717364483, 3.73867544391172550624435727116, 4.863453411308963272441903498666, 5.53827817070936624371958816376, 6.49197907774742537829489244936, 7.11017426019027409543438333124, 8.05088429844749959614130056547, 9.05654484108839479882747282231, 9.47985834590449126637744632933, 10.31333026330149345727521970212, 11.25961978701498371499349877234, 12.11877990048138489317542776002, 12.51443139215805413053158470132, 13.4509873430111571667971391490, 14.35096222856582342018724876462, 14.7930882668165736795161295660, 15.94538958923402261431821122381, 16.29517802839345777737652697045, 17.02171864386561737743810475540, 18.05023398704650645214414384134, 18.76799102204260111686160102858, 19.13106996163554080671901308961, 20.05253333806784798816330139389

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