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Properties

Label 1-1080-1080.77-r0-0-0
Degree $1$
Conductor $1080$
Sign $-0.574 + 0.818i$
Analytic cond. $5.01549$
Root an. cond. $5.01549$
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  + (0.984 + 0.173i)7-s + (−0.939 + 0.342i)11-s + (0.642 + 0.766i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.984 + 0.173i)23-s + (−0.766 − 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.939 + 0.342i)49-s − i·53-s + (0.939 + 0.342i)59-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)7-s + (−0.939 + 0.342i)11-s + (0.642 + 0.766i)13-s + (−0.866 + 0.5i)17-s + (−0.5 + 0.866i)19-s + (−0.984 + 0.173i)23-s + (−0.766 − 0.642i)29-s + (0.173 + 0.984i)31-s + (−0.866 + 0.5i)37-s + (−0.766 + 0.642i)41-s + (−0.342 − 0.939i)43-s + (−0.984 − 0.173i)47-s + (0.939 + 0.342i)49-s − i·53-s + (0.939 + 0.342i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1080\)    =    \(2^{3} \cdot 3^{3} \cdot 5\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(5.01549\)
Root analytic conductor: \(5.01549\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1080} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1080,\ (0:\ ),\ -0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4261543715 + 0.8195218892i\)
\(L(\frac12)\) \(\approx\) \(0.4261543715 + 0.8195218892i\)
\(L(1)\) \(\approx\) \(0.9155916285 + 0.2292461364i\)
\(L(1)\) \(\approx\) \(0.9155916285 + 0.2292461364i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.984 + 0.173i)T \)
11 \( 1 + (-0.939 + 0.342i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (-0.642 - 0.766i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (0.642 - 0.766i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−21.0559782915557162264224379517, −20.49442485794198141408105080566, −19.81185882253847418408318789054, −18.66270162711498565532290587104, −18.038084947612745882500259304751, −17.51521443647004252845102680438, −16.4725963950019709897457526715, −15.600354925047840149048077702689, −15.08363535634304472757856912630, −13.995874643965008558963790608568, −13.38155572179822543000429634945, −12.62031106088934847479769263892, −11.40044415608309263638754324315, −10.97080506817487600928209569384, −10.18189380183551356248021870490, −8.99095587385097713051799304569, −8.24416302710850244823754160769, −7.59701757723854511517490226513, −6.536113023561742613597302879360, −5.495756221827643955997690072686, −4.814011675703427229991405478151, −3.81060900247914994368786850895, −2.66976341109024798627708942539, −1.76736976618122811221094245758, −0.353986049796785894937300324401, 1.65223819177574660801591325991, 2.13135861189079249248222287550, 3.59634678481862846373586267893, 4.45209072869018626566594691088, 5.302615739147761522627000519053, 6.23504644848430134204325831825, 7.18844433373675661748059426149, 8.288844706085237494511013398681, 8.55336481596637447216949214404, 9.89019270465684437516735846130, 10.60173697471712311188927350662, 11.44314958147162713755611845983, 12.136284598136776840838796865567, 13.16616646246763547773586888211, 13.843557951531312903067762623693, 14.74911980389141041422509292897, 15.418546689635109193429489105074, 16.22782712168615401785211474425, 17.14347398333397045272451808893, 17.96370898569207465047688465645, 18.47153090579983389655469077695, 19.34919054898985751055048738137, 20.35634186938890315336675051471, 21.00121377806889817454920258301, 21.51733083580796210005917770703

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