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Properties

Label 2-45-5.4-c17-0-8
Degree $2$
Conductor $45$
Sign $-0.650 - 0.759i$
Analytic cond. $82.4499$
Root an. cond. $9.08019$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 507. i·2-s − 1.26e5·4-s + (−5.67e5 − 6.63e5i)5-s − 7.16e6i·7-s + 2.38e6i·8-s + (3.36e8 − 2.88e8i)10-s − 1.34e9·11-s − 4.11e8i·13-s + 3.63e9·14-s − 1.77e10·16-s − 3.37e10i·17-s + 4.51e10·19-s + (7.17e10 + 8.38e10i)20-s − 6.80e11i·22-s − 4.67e11i·23-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.964·4-s + (−0.650 − 0.759i)5-s − 0.470i·7-s + 0.0502i·8-s + (1.06 − 0.911i)10-s − 1.88·11-s − 0.139i·13-s + 0.658·14-s − 1.03·16-s − 1.17i·17-s + 0.609·19-s + (0.626 + 0.732i)20-s − 2.64i·22-s − 1.24i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.650 - 0.759i$
Analytic conductor: \(82.4499\)
Root analytic conductor: \(9.08019\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :17/2),\ -0.650 - 0.759i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.9326689930\)
\(L(\frac12)\) \(\approx\) \(0.9326689930\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (5.67e5 + 6.63e5i)T \)
good2 \( 1 - 507. iT - 1.31e5T^{2} \)
7 \( 1 + 7.16e6iT - 2.32e14T^{2} \)
11 \( 1 + 1.34e9T + 5.05e17T^{2} \)
13 \( 1 + 4.11e8iT - 8.65e18T^{2} \)
17 \( 1 + 3.37e10iT - 8.27e20T^{2} \)
19 \( 1 - 4.51e10T + 5.48e21T^{2} \)
23 \( 1 + 4.67e11iT - 1.41e23T^{2} \)
29 \( 1 - 3.39e12T + 7.25e24T^{2} \)
31 \( 1 + 1.73e12T + 2.25e25T^{2} \)
37 \( 1 - 3.71e13iT - 4.56e26T^{2} \)
41 \( 1 + 9.02e13T + 2.61e27T^{2} \)
43 \( 1 - 4.31e13iT - 5.87e27T^{2} \)
47 \( 1 - 1.79e14iT - 2.66e28T^{2} \)
53 \( 1 + 2.79e14iT - 2.05e29T^{2} \)
59 \( 1 - 1.18e15T + 1.27e30T^{2} \)
61 \( 1 - 1.54e15T + 2.24e30T^{2} \)
67 \( 1 - 2.66e15iT - 1.10e31T^{2} \)
71 \( 1 + 1.19e14T + 2.96e31T^{2} \)
73 \( 1 + 4.39e15iT - 4.74e31T^{2} \)
79 \( 1 + 9.50e15T + 1.81e32T^{2} \)
83 \( 1 - 5.86e15iT - 4.21e32T^{2} \)
89 \( 1 + 2.12e16T + 1.37e33T^{2} \)
97 \( 1 - 4.45e16iT - 5.95e33T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.00686239978660318226490240815, −11.60266331858401219377705151790, −10.15041170452499774311132964638, −8.529117414958977678612714611306, −7.84849633407013995214876847303, −6.87268853205080035987317005753, −5.27051895754792741290342511513, −4.69045717231260575227502513532, −2.79212285119784438024783974073, −0.67973813668015822993106390967, 0.32499441906161293623299747205, 1.95701526688779948247194905298, 2.88584541635266592227004384505, 3.83430907973752644787947132546, 5.42799810260087969529049609306, 7.17948336664945837060658388111, 8.432885389520064410346702853607, 10.05678552094057790360073979384, 10.71672770396489086866672564826, 11.71354092480053656098389712675

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