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Properties

Label 1-1980-1980.1183-r1-0-0
Degree $1$
Conductor $1980$
Sign $-0.936 + 0.349i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1980 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1980\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 11\)
Sign: $-0.936 + 0.349i$
Analytic conductor: \(212.780\)
Root analytic conductor: \(212.780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1980} (1183, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1980,\ (1:\ ),\ -0.936 + 0.349i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1042095637 - 0.5775012979i\)
\(L(\frac12)\) \(\approx\) \(-0.1042095637 - 0.5775012979i\)
\(L(1)\) \(\approx\) \(0.9777654124 - 0.2376229034i\)
\(L(1)\) \(\approx\) \(0.9777654124 - 0.2376229034i\)

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

−20.36253342967217659229156320820, −19.26231785376184851116246512206, −18.75480975221102324140043483819, −18.13209226698663152635866830801, −17.25175010537297125233350078074, −16.616422955083245896108693510714, −15.782916186889577946120186944614, −15.06705790519908671745959855442, −14.23243050070790414665362743542, −13.88709308772640534357970295145, −12.54616367338687012966132767533, −12.19367498046074151791624860069, −11.35179351113029499618868630613, −10.56830797386358802144022949340, −9.801984190957007744800290788881, −8.78415018729458809207196055823, −8.29883452765625727798486151444, −7.51246324654907751890820490223, −6.33789642771365800326306285221, −5.86818390581774685142117379603, −4.83282817652268475172609145436, −4.080722753407132024230256161700, −3.15823697093947398776262882803, −1.898217207333370908171240893332, −1.5258528467441528273127733310, 0.10161846678503073477122605754, 1.04255087331069224021149260107, 1.97598709990300447984367416190, 3.13116138979400239531511147850, 3.850711638093257157895161699094, 4.94553608142290547605050723237, 5.40471273203309981191772908348, 6.58602049154407652848536963994, 7.34009722941171091265167060683, 8.05633580410136062836575256154, 8.77030066956611424837770481112, 9.79468517483166838418791615620, 10.49787456330443846698630465406, 11.18394976133433385014391813230, 11.90774084771910315408063347, 12.804612314219838798095206408121, 13.64673800762315518560454310711, 14.113740231891028132726842574806, 15.05208324307044781643832253904, 15.668254859669780060813196624475, 16.56262025982976868339853619981, 17.21376512921262608115666467955, 18.08905979358669586108638161845, 18.3412411769404233824273024509, 19.63730663486817303177653558962

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