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Properties

Label 2-45-9.4-c3-0-3
Degree $2$
Conductor $45$
Sign $0.891 - 0.452i$
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  + (0.112 + 0.195i)2-s + (2.06 + 4.76i)3-s + (3.97 − 6.88i)4-s + (2.5 − 4.33i)5-s + (−0.697 + 0.940i)6-s + (15.5 + 27.0i)7-s + 3.59·8-s + (−18.4 + 19.7i)9-s + 1.12·10-s + (−9.06 − 15.6i)11-s + (41.0 + 4.71i)12-s + (25.0 − 43.4i)13-s + (−3.51 + 6.08i)14-s + (25.8 + 2.96i)15-s + (−31.3 − 54.3i)16-s − 131.·17-s + ⋯
L(s)  = 1  + (0.0398 + 0.0689i)2-s + (0.397 + 0.917i)3-s + (0.496 − 0.860i)4-s + (0.223 − 0.387i)5-s + (−0.0474 + 0.0639i)6-s + (0.842 + 1.45i)7-s + 0.158·8-s + (−0.683 + 0.730i)9-s + 0.0356·10-s + (−0.248 − 0.430i)11-s + (0.987 + 0.113i)12-s + (0.535 − 0.926i)13-s + (−0.0670 + 0.116i)14-s + (0.444 + 0.0510i)15-s + (−0.490 − 0.849i)16-s − 1.87·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.891 - 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(45\)    =    \(3^{2} \cdot 5\)
Sign: $0.891 - 0.452i$
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: \((2,\ 45,\ (\ :3/2),\ 0.891 - 0.452i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.66901 + 0.399059i\)
\(L(\frac12)\) \(\approx\) \(1.66901 + 0.399059i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.06 - 4.76i)T \)
5 \( 1 + (-2.5 + 4.33i)T \)
good2 \( 1 + (-0.112 - 0.195i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-15.5 - 27.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (9.06 + 15.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-25.0 + 43.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 131.T + 4.91e3T^{2} \)
19 \( 1 - 23.2T + 6.85e3T^{2} \)
23 \( 1 + (-16.4 + 28.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (62.9 + 108. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (62.5 - 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 99.9T + 5.06e4T^{2} \)
41 \( 1 + (122. - 212. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (69.5 + 120. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-236. - 409. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 421.T + 1.48e5T^{2} \)
59 \( 1 + (-371. + 642. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (4.48 + 7.77i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (294. - 510. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 48.5T + 3.57e5T^{2} \)
73 \( 1 - 409.T + 3.89e5T^{2} \)
79 \( 1 + (265. + 459. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-147. - 255. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 852.T + 7.04e5T^{2} \)
97 \( 1 + (-194. - 336. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.43684900009218097353940449693, −14.66294642660633593157111571721, −13.34889027848461253241925396044, −11.52393584056977093216446894442, −10.69777433925152051800288582124, −9.243553547710721295575398106965, −8.341813061030372920134264071465, −5.90026662740230798438607486753, −4.94361538956900600943908722609, −2.37663147063899195611969337229, 1.99445867757048557049713141775, 3.99983435942350810613332921568, 6.80566319426963317090224096213, 7.41877317136620733971168837328, 8.760080832775206845865795983812, 10.84404799396608768308040733543, 11.65971910168347681489592219358, 13.23194581337909200249275273311, 13.74061364923431661400291784525, 15.07403488587023899191953373901

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