Properties

Degree 1
Conductor $ 2^{3} \cdot 3^{2} $
Sign $-0.642 + 0.766i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + 35-s − 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯
L(s,χ)  = 1  + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)13-s − 17-s − 19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 + 0.866i)29-s + (−0.5 − 0.866i)31-s + 35-s − 37-s + (0.5 + 0.866i)41-s + (0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.642 + 0.766i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 72 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.642 + 0.766i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(72\)    =    \(2^{3} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.642 + 0.766i$
motivic weight  =  \(0\)
character  :  $\chi_{72} (5, \cdot )$
Sato-Tate  :  $\mu(6)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 72,\ (1:\ ),\ -0.642 + 0.766i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.2464516952 + 0.5285173660i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.2464516952 + 0.5285173660i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7292309272 + 0.1285830876i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7292309272 + 0.1285830876i\)

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

−30.83037668619064299373213638249, −29.9253246644374674658240703520, −29.01728381798350693365328633522, −27.4980391174521508955933955600, −26.57978779530438895528603755859, −25.82188306221973995012455746989, −24.27252356467162525837197831430, −23.1356160962578843158485408964, −22.433788639640670503583600643966, −20.9956782784547701349659788782, −19.72543107989759407762863886028, −18.83605536408663314060370627049, −17.599868931216853304629813732149, −16.21491733750108355511825043979, −15.21406030141763755971769573511, −13.86301526922421918116580714653, −12.80483613200410822700470834326, −10.982045826259326076251787980679, −10.49560746384215428801756113553, −8.58456323433546704586149691284, −7.272743079556181740697003556942, −6.10716495282425505885013601827, −4.09044499945765741014199177242, −2.85360662224328524951619800970, −0.27916258375206402860741786269, 2.03050722491665800089732416894, 4.01622104871305997660880673099, 5.359351407905197843229220904511, 6.952061741311805215844667049271, 8.55848332628303009177429000596, 9.41394751370787855309455340901, 11.193696365929633450979002541091, 12.41787325712974320205590692161, 13.248916306512317853321177030475, 15.116958560494163518727844529694, 15.888019408777505971879821825566, 17.10178493776695212447335825768, 18.50314773245295014463399304073, 19.56557329187023301823663626792, 20.68913660127655506202593361556, 21.77551930685614350873765997917, 23.13319123428032980961110407147, 24.05080278794069172920281652906, 25.24367831979855411211771531534, 26.18311224488381485006889801226, 27.75110406351776935699382474693, 28.36227190348329599162538758019, 29.36158464739641785688926857959, 31.261379570712443886354722830929, 31.33816765057265948657028144066

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