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Properties

Label 2-45-9.4-c3-0-7
Degree $2$
Conductor $45$
Sign $-0.457 + 0.888i$
Analytic cond. $2.65508$
Root an. cond. $1.62944$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.785 − 1.35i)2-s + (−3.89 + 3.43i)3-s + (2.76 − 4.79i)4-s + (2.5 − 4.33i)5-s + (7.73 + 2.60i)6-s + (−17.1 − 29.6i)7-s − 21.2·8-s + (3.40 − 26.7i)9-s − 7.85·10-s + (13.4 + 23.2i)11-s + (5.67 + 28.1i)12-s + (9.74 − 16.8i)13-s + (−26.8 + 46.5i)14-s + (5.12 + 25.4i)15-s + (−5.45 − 9.44i)16-s + 29.1·17-s + ⋯
L(s)  = 1  + (−0.277 − 0.480i)2-s + (−0.750 + 0.660i)3-s + (0.345 − 0.599i)4-s + (0.223 − 0.387i)5-s + (0.526 + 0.177i)6-s + (−0.924 − 1.60i)7-s − 0.939·8-s + (0.126 − 0.991i)9-s − 0.248·10-s + (0.368 + 0.638i)11-s + (0.136 + 0.678i)12-s + (0.207 − 0.360i)13-s + (−0.513 + 0.888i)14-s + (0.0881 + 0.438i)15-s + (−0.0852 − 0.147i)16-s + 0.415·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $-0.457 + 0.888i$
Analytic conductor: \(2.65508\)
Root analytic conductor: \(1.62944\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{45} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 45,\ (\ :3/2),\ -0.457 + 0.888i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.422147 - 0.692335i\)
\(L(\frac12)\) \(\approx\) \(0.422147 - 0.692335i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.89 - 3.43i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (0.785 + 1.35i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (17.1 + 29.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-13.4 - 23.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-9.74 + 16.8i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 29.1T + 4.91e3T^{2} \)
19 \( 1 - 47.1T + 6.85e3T^{2} \)
23 \( 1 + (56.6 - 98.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (40.6 + 70.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-5.41 + 9.38i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 410.T + 5.06e4T^{2} \)
41 \( 1 + (-221. + 384. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (169. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (118. + 204. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 609.T + 1.48e5T^{2} \)
59 \( 1 + (7.77 - 13.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (108. - 187. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 65.4T + 3.57e5T^{2} \)
73 \( 1 - 711.T + 3.89e5T^{2} \)
79 \( 1 + (-478. - 829. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-261. - 452. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 1.60e3T + 7.04e5T^{2} \)
97 \( 1 + (400. + 694. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.20215229464538896773751174461, −13.75410195563793967698397066506, −12.35705787504070186455378428848, −11.12712724899072013999526021198, −10.03855325929038508499297055836, −9.600338725766489915007324599962, −7.04309268389863819486002712869, −5.70654165894178702851481590302, −3.86645805288372602250674808013, −0.77666231863224725642460879020, 2.76439527425816308893529072094, 5.90893774141052731737060652027, 6.52758037185372778027171707761, 8.093922131081465246141069652788, 9.421431938792019109440109506477, 11.34607960720252454230216106954, 12.16053614097270707899188964049, 13.10431849313761856917368892372, 14.78710216941381956365343807310, 16.20212988312068500251338286417

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