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Properties

Label 1-1380-1380.167-r1-0-0
Degree $1$
Conductor $1380$
Sign $-0.408 - 0.912i$
Analytic cond. $148.301$
Root an. cond. $148.301$
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  + (−0.281 + 0.959i)7-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (−0.989 + 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.909 + 0.415i)37-s + (−0.415 + 0.909i)41-s + (−0.540 + 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (0.281 − 0.959i)53-s + (0.959 − 0.281i)59-s + (0.841 − 0.540i)61-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)7-s + (−0.654 + 0.755i)11-s + (0.281 + 0.959i)13-s + (−0.989 + 0.142i)17-s + (−0.142 + 0.989i)19-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (−0.909 + 0.415i)37-s + (−0.415 + 0.909i)41-s + (−0.540 + 0.841i)43-s − i·47-s + (−0.841 − 0.540i)49-s + (0.281 − 0.959i)53-s + (0.959 − 0.281i)59-s + (0.841 − 0.540i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.408 - 0.912i$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ -0.408 - 0.912i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1884293125 + 0.2908189582i\)
\(L(\frac12)\) \(\approx\) \(-0.1884293125 + 0.2908189582i\)
\(L(1)\) \(\approx\) \(0.7652960203 + 0.2849658963i\)
\(L(1)\) \(\approx\) \(0.7652960203 + 0.2849658963i\)

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 \( 1 + (-0.281 + 0.959i)T \)
11 \( 1 + (-0.654 + 0.755i)T \)
13 \( 1 + (0.281 + 0.959i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.909 + 0.415i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.540 + 0.841i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.281 - 0.959i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (-0.755 + 0.654i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.989 + 0.142i)T \)
79 \( 1 + (-0.959 + 0.281i)T \)
83 \( 1 + (0.909 - 0.415i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
97 \( 1 + (0.909 + 0.415i)T \)
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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

−20.17790322028133348389956427307, −19.506171539530428207094272265808, −18.574124908975432369158977252249, −17.830135789534488440146950878240, −17.14680931476182523569583306611, −16.27490468728837924298361817162, −15.654433669303923698229840017203, −14.88154978213418324100706673125, −13.71445219458539684038207539210, −13.37002132154907445114585226559, −12.65765004593733712413969211234, −11.45806500407663633498426047860, −10.67793310638181330437137987012, −10.33871231447033694410348258291, −9.057686050471549670938361364436, −8.46835240766554527810705248245, −7.376237702207768052869882706273, −6.858980467043518793163643367485, −5.72633668866502590732103659322, −4.99536380951214524622037891289, −3.876122340113561634788799690823, −3.17827744053922893678995706907, −2.12195248279805617046432465956, −0.73506961814179978108731315595, −0.084934901693429366641216054536, 1.68661167705166637520251832417, 2.27017110336839193478510894591, 3.39094424604510636266228549540, 4.410939116500672714880600094653, 5.22948560118834158021929610372, 6.20376518532645821634001527755, 6.84343458576597194290005497571, 7.957491285088812433201669927912, 8.690801667500221603349421585404, 9.51043547858628639641805882400, 10.205422915810769974998507093714, 11.26776931062453045464409982296, 11.9239103546335630441758228294, 12.767851138395465308749534692598, 13.35532842995446267797196670106, 14.45192546941003091760635893907, 15.08672450531023937989792625504, 15.8789150574835171004796645038, 16.44397796410012365405392080607, 17.53445091137915225474611513473, 18.17731611401586092572322621144, 18.885899218843326379896274889261, 19.51334298196861105083819593791, 20.554926179929313961325082658799, 21.116915810109838837138512349490

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