Properties

Degree 1
Conductor $ 3^{2} \cdot 5 $
Sign $0.929 + 0.370i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s i·17-s − 19-s + (−0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s − 26-s i·28-s + (−0.5 − 0.866i)29-s + ⋯
L(s,χ)  = 1  + (−0.866 + 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 − 0.5i)7-s + i·8-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s i·17-s − 19-s + (−0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s − 26-s i·28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.929 + 0.370i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.929 + 0.370i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(45\)    =    \(3^{2} \cdot 5\)
\( \varepsilon \)  =  $0.929 + 0.370i$
motivic weight  =  \(0\)
character  :  $\chi_{45} (2, \cdot )$
Sato-Tate  :  $\mu(12)$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 45,\ (0:\ ),\ 0.929 + 0.370i)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(0.6186750856 + 0.1186760323i\)
\(L(\frac12,\chi)\)  \(\approx\)  \(0.6186750856 + 0.1186760323i\)
\(L(\chi,1)\)  \(\approx\)  \(0.7440027170 + 0.1180152635i\)
\(L(1,\chi)\)  \(\approx\)  \(0.7440027170 + 0.1180152635i\)

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

−34.62967785083970788898507180291, −33.38979987193827460187196568341, −31.75497866774752961726753609955, −30.446674166988685084712748527400, −29.69894652154906483824578189733, −28.10309283478446486624817585952, −27.606948248345891642137748355420, −26.22684287698595384304202209410, −25.11132232099705185782462287886, −23.89347784692159102147371808229, −21.94044422227897593488232921148, −21.116293649872633667062894048396, −19.79822850542316345509697109064, −18.61109460302128361749755597333, −17.5921211914564113937988371692, −16.34204058802709121944223434347, −14.869705467444748609823099402970, −13.040344954552693888755750638701, −11.58116042577649636338356502441, −10.64290908413794165123591939364, −8.8916162329185974858423703850, −8.01638507168060770541567812440, −6.06092527099599308369164858061, −3.71393660200290484575758359195, −1.75276955785323066086558323353, 1.73502892654605288461993531888, 4.578501693340769035221711208043, 6.448851107830590197365472259357, 7.7476119182222896718349891881, 9.07333274769736290563340464442, 10.507194128477480217418474459614, 11.73149830143295670844100560315, 13.909287013660276397081441892307, 15.01253481041527087360323726737, 16.417230031553354474935915367, 17.5345942325096034319351793937, 18.52301522851174264287221231315, 19.99348626087132115415923380481, 20.94589780967766167649156688144, 22.957280520520294536518072905453, 23.99866204216137566852635920107, 25.169759660374015048257708100377, 26.245563648684268132081691340665, 27.44603780745807575770257362501, 28.21848491514406267014544797250, 29.65893824571317793003524846613, 30.80609585445922773383117873663, 32.50116374330450786624458952740, 33.5627824516012812434326858147, 34.27319596443480332218445129009

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