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Properties

Label 1-45-45.29-r1-0-0
Degree $1$
Conductor $45$
Sign $0.984 - 0.173i$
Analytic cond. $4.83592$
Root an. cond. $4.83592$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + 19-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 26-s − 28-s + (0.5 + 0.866i)29-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)7-s + 8-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s + 19-s + (0.5 − 0.866i)22-s + (−0.5 + 0.866i)23-s − 26-s − 28-s + (0.5 + 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(4.83592\)
Root analytic conductor: \(4.83592\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 45,\ (1:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.206579964 - 0.1055620685i\)
\(L(\frac12)\) \(\approx\) \(1.206579964 - 0.1055620685i\)
\(L(1)\) \(\approx\) \(0.9224122680 - 0.1626461701i\)
\(L(1)\) \(\approx\) \(0.9224122680 - 0.1626461701i\)

Euler product

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

−34.09354844432550190639116750119, −33.096614944814669778618110813928, −32.169502865313291286750099835899, −30.685499806272144250845357844342, −29.29842349904856612876547413058, −27.960650724316858179136570486586, −26.881607302044681586248722288725, −26.062727534599654330717055294747, −24.58099736284453522983527464648, −23.78929074870833939877883357147, −22.56747582479688063158985794977, −20.873706800250642307277526773646, −19.41763203336929951676347582497, −18.29335636989591916245291879947, −16.93965815091351875558743929500, −16.15735221060706352418911039008, −14.426178629447459084940200036191, −13.71176218491971948402683198965, −11.44543758704392776439917913229, −10.04172275714930063405894550472, −8.57948695826822989436992851989, −7.3181106414903039163295544906, −5.88676066227881484131819960130, −4.147606581579310469065204414760, −1.067351993257728218488609668356, 1.54474594651549325427733735412, 3.330298957431496527835844980089, 5.24499814264478250614496320864, 7.58742421616382402395643119103, 8.919514238728154640911584339105, 10.19123787481185370580701801517, 11.68483267851063612348286288548, 12.58558247617182811327977389699, 14.26802439262418854060186318201, 15.88705110834848360153635660097, 17.57346243938264112789622021261, 18.310427350710028233853591129309, 19.71887573843978783522119987479, 20.79013162927799356246445393484, 21.91774727990832234613887797398, 23.05808537763083650546840655941, 24.94602908369519069118581505012, 25.87797159072455967631067350577, 27.56947017759310200020462943649, 27.94554490147556513983343776205, 29.35050010777095582773298479366, 30.528876205286991057218563581165, 31.30280267583153958988684296616, 32.745843343968825180805275378941, 34.381399504681706271077370772116

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