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Properties

Label 2-15e2-45.4-c3-0-21
Degree $2$
Conductor $225$
Sign $0.489 + 0.871i$
Analytic cond. $13.2754$
Root an. cond. $3.64354$
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  + (−1.35 + 0.785i)2-s + (−3.43 − 3.89i)3-s + (−2.76 + 4.79i)4-s + (7.73 + 2.60i)6-s + (−29.6 + 17.1i)7-s − 21.2i·8-s + (−3.40 + 26.7i)9-s + (13.4 + 23.2i)11-s + (28.1 − 5.67i)12-s + (16.8 + 9.74i)13-s + (26.8 − 46.5i)14-s + (−5.45 − 9.44i)16-s − 29.1i·17-s + (−16.3 − 39.0i)18-s − 47.1·19-s + ⋯
L(s)  = 1  + (−0.480 + 0.277i)2-s + (−0.660 − 0.750i)3-s + (−0.345 + 0.599i)4-s + (0.526 + 0.177i)6-s + (−1.60 + 0.924i)7-s − 0.939i·8-s + (−0.126 + 0.991i)9-s + (0.368 + 0.638i)11-s + (0.678 − 0.136i)12-s + (0.360 + 0.207i)13-s + (0.513 − 0.888i)14-s + (−0.0852 − 0.147i)16-s − 0.415i·17-s + (−0.214 − 0.511i)18-s − 0.568·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.489 + 0.871i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.319114 - 0.186811i\)
\(L(\frac12)\) \(\approx\) \(0.319114 - 0.186811i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.10260453671829058736987724072, −10.58053459080056808745611214495, −9.428325712854648797655159617075, −8.763775990904011843232433063872, −7.45312675056202773239736212655, −6.64340246706984705568597304861, −5.78427587947936671493627124748, −4.05550445837687172155148228845, −2.47337611119762628965611681673, −0.27765056299144310366469492778, 0.869741512564533890107273037453, 3.35638755222441200216559424074, 4.42722811053421869181910973840, 5.96576927554438838432890371966, 6.47401704948696370814517660347, 8.309215680711942954106507336787, 9.517563827787238224807319924048, 9.966619542463916695920821032214, 10.73801776765432361709380698654, 11.59712805511262439657255250536

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