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Properties

Label 2-45-5.4-c9-0-6
Degree $2$
Conductor $45$
Sign $-0.406 - 0.913i$
Analytic cond. $23.1766$
Root an. cond. $4.81420$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.843i·2-s + 511.·4-s + (568. + 1.27e3i)5-s + 8.71e3i·7-s + 863. i·8-s + (−1.07e3 + 479. i)10-s − 4.45e4·11-s + 2.14e4i·13-s − 7.35e3·14-s + 2.61e5·16-s − 3.00e5i·17-s − 5.65e5·19-s + (2.90e5 + 6.52e5i)20-s − 3.76e4i·22-s + 9.50e5i·23-s + ⋯
L(s)  = 1  + 0.0372i·2-s + 0.998·4-s + (0.406 + 0.913i)5-s + 1.37i·7-s + 0.0745i·8-s + (−0.0340 + 0.0151i)10-s − 0.917·11-s + 0.208i·13-s − 0.0511·14-s + 0.995·16-s − 0.871i·17-s − 0.995·19-s + (0.406 + 0.912i)20-s − 0.0342i·22-s + 0.708i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.406 - 0.913i$
Analytic conductor: \(23.1766\)
Root analytic conductor: \(4.81420\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :9/2),\ -0.406 - 0.913i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.15773 + 1.78252i\)
\(L(\frac12)\) \(\approx\) \(1.15773 + 1.78252i\)
\(L(\frac{11}{2})\) 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 \( 1 + (-568. - 1.27e3i)T \)
good2 \( 1 - 0.843iT - 512T^{2} \)
7 \( 1 - 8.71e3iT - 4.03e7T^{2} \)
11 \( 1 + 4.45e4T + 2.35e9T^{2} \)
13 \( 1 - 2.14e4iT - 1.06e10T^{2} \)
17 \( 1 + 3.00e5iT - 1.18e11T^{2} \)
19 \( 1 + 5.65e5T + 3.22e11T^{2} \)
23 \( 1 - 9.50e5iT - 1.80e12T^{2} \)
29 \( 1 + 8.03e5T + 1.45e13T^{2} \)
31 \( 1 + 1.99e6T + 2.64e13T^{2} \)
37 \( 1 - 9.53e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.54e7T + 3.27e14T^{2} \)
43 \( 1 - 2.32e7iT - 5.02e14T^{2} \)
47 \( 1 - 3.77e7iT - 1.11e15T^{2} \)
53 \( 1 + 4.79e7iT - 3.29e15T^{2} \)
59 \( 1 - 7.00e7T + 8.66e15T^{2} \)
61 \( 1 - 1.26e8T + 1.16e16T^{2} \)
67 \( 1 + 2.66e8iT - 2.72e16T^{2} \)
71 \( 1 + 6.59e7T + 4.58e16T^{2} \)
73 \( 1 + 1.47e7iT - 5.88e16T^{2} \)
79 \( 1 - 4.66e7T + 1.19e17T^{2} \)
83 \( 1 - 2.01e8iT - 1.86e17T^{2} \)
89 \( 1 - 5.54e8T + 3.50e17T^{2} \)
97 \( 1 - 3.39e8iT - 7.60e17T^{2} \)
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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

−14.52816506861330869768739001350, −12.95760531576800182306032412104, −11.69592140733734252548195597554, −10.82069147066935823973499443201, −9.503847103367752762416587986939, −7.84695280548816445593486160554, −6.52646129821853441478236262035, −5.49157840044399043533756646601, −2.91191379957126499927687700895, −2.09992988791923804794318287799, 0.66820303089539268786698504141, 2.12796389799797064857445395852, 4.07647741057103887809356520821, 5.74411179177881503468225659282, 7.17568885638856124330146944124, 8.366538126190326081450348427014, 10.19883337046759377006306565737, 10.85578205116889038538357689559, 12.50640487937734867598359794339, 13.26821210320160742029531529556

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