Properties

Degree 1
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.559 - 0.828i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)11-s + (0.654 − 0.755i)13-s + (0.841 − 0.540i)17-s + (0.959 − 0.281i)19-s + (0.142 − 0.989i)23-s − 29-s + (0.654 + 0.755i)31-s − 37-s + (−0.841 + 0.540i)41-s + (0.841 − 0.540i)43-s + (0.142 − 0.989i)47-s + (0.841 + 0.540i)49-s + (0.841 + 0.540i)53-s + (−0.654 − 0.755i)59-s + ⋯
L(s,χ)  = 1  + (−0.959 − 0.281i)7-s + (0.415 + 0.909i)11-s + (0.654 − 0.755i)13-s + (0.841 − 0.540i)17-s + (0.959 − 0.281i)19-s + (0.142 − 0.989i)23-s − 29-s + (0.654 + 0.755i)31-s − 37-s + (−0.841 + 0.540i)41-s + (0.841 − 0.540i)43-s + (0.142 − 0.989i)47-s + (0.841 + 0.540i)49-s + (0.841 + 0.540i)53-s + (−0.654 − 0.755i)59-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.559 - 0.828i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.559 - 0.828i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.559 - 0.828i$
motivic weight  =  \(0\)
character  :  $\chi_{4020} (59, \cdot )$
Sato-Tate  :  $\mu(22)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 4020,\ (0:\ ),\ 0.559 - 0.828i)$
$L(\chi,\frac{1}{2})$  $\approx$  $1.387851800 - 0.7371293443i$
$L(\frac12,\chi)$  $\approx$  $1.387851800 - 0.7371293443i$
$L(\chi,1)$  $\approx$  1.046274458 - 0.1283899169i
$L(1,\chi)$  $\approx$  1.046274458 - 0.1283899169i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−18.77098519915282599086505544184, −18.0131095450369542110003810369, −16.964811044764535006726729371557, −16.60470425984265431213251952364, −15.90052237566664391260507121682, −15.33185073627072077743005374402, −14.4018407769425124201239575363, −13.692415449062070751736364444139, −13.30244029640732526132552536902, −12.301880464542288674663241042840, −11.777290001606603547572157928020, −11.08889865766770472456794228283, −10.2332060054955251851647893155, −9.448531440654940834459402664065, −9.03239294852916052429684922149, −8.15625530757431363559402436701, −7.35880737404305372584667016537, −6.54521640854017348649232785191, −5.83144736306714482383220275043, −5.42514130158056164531423111323, −4.02823785903330558267647596822, −3.56579227449297815034956775810, −2.89182828435596269965453938581, −1.7172565312832134448445845126, −0.927673377980015775269483103923, 0.5503928767408337577780777050, 1.42749147459843627540941288038, 2.57635906227646514914968348995, 3.328918430277604571282842966655, 3.90823292871422123000470653076, 4.981502079209416595614807056572, 5.5839306918278230963890058045, 6.56720830714521802523391642710, 7.059853417774514920600368480203, 7.79709413775917413910843397977, 8.727895598050960858959467821328, 9.43779472501723460955645106462, 10.091043698500393896154318732, 10.59140790296691832224147251748, 11.610496247349328185184779568495, 12.30820648914581664285861776341, 12.80623126374445376897256079948, 13.641747112662192780005021445408, 14.16176318496708245626437270299, 15.09838504731307855778154759240, 15.674336922670863643304016848086, 16.30932625895576583569753749577, 16.99460783735246497360415395391, 17.65154262162702915981557411068, 18.49615579498918929751159388270

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