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Properties

Label 1-45-45.2-r0-0-0
Degree $1$
Conductor $45$
Sign $0.929 + 0.370i$
Analytic cond. $0.208979$
Root an. cond. $0.208979$
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  + (−0.866 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.929 + 0.370i$
Analytic conductor: \(0.208979\)
Root analytic conductor: \(0.208979\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 45,\ (0:\ ),\ 0.929 + 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6186750856 + 0.1186760323i\)
\(L(\frac12)\) \(\approx\) \(0.6186750856 + 0.1186760323i\)
\(L(1)\) \(\approx\) \(0.7440027170 + 0.1180152635i\)
\(L(1)\) \(\approx\) \(0.7440027170 + 0.1180152635i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 - T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−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