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Properties

Label 2-15e2-45.34-c3-0-1
Degree $2$
Conductor $225$
Sign $-0.956 - 0.292i$
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  + (2.05 + 1.18i)2-s + (−4.97 − 1.5i)3-s + (−1.18 − 2.05i)4-s + (−8.44 − 8.98i)6-s + (6.38 + 3.68i)7-s − 24.6i·8-s + (22.5 + 14.9i)9-s + (2.06 − 3.58i)11-s + (2.81 + 11.9i)12-s + (−68.3 + 39.4i)13-s + (8.74 + 15.1i)14-s + (19.6 − 34.1i)16-s + 33.3i·17-s + (28.5 + 57.3i)18-s − 89.3·19-s + ⋯
L(s)  = 1  + (0.726 + 0.419i)2-s + (−0.957 − 0.288i)3-s + (−0.148 − 0.256i)4-s + (−0.574 − 0.611i)6-s + (0.344 + 0.199i)7-s − 1.08i·8-s + (0.833 + 0.552i)9-s + (0.0567 − 0.0982i)11-s + (0.0678 + 0.288i)12-s + (−1.45 + 0.842i)13-s + (0.166 + 0.289i)14-s + (0.307 − 0.533i)16-s + 0.475i·17-s + (0.373 + 0.750i)18-s − 1.07·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.956 - 0.292i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.956 - 0.292i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0350932 + 0.234922i\)
\(L(\frac12)\) \(\approx\) \(0.0350932 + 0.234922i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.97 + 1.5i)T \)
5 \( 1 \)
good2 \( 1 + (-2.05 - 1.18i)T + (4 + 6.92i)T^{2} \)
7 \( 1 + (-6.38 - 3.68i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-2.06 + 3.58i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (68.3 - 39.4i)T + (1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 33.3iT - 4.91e3T^{2} \)
19 \( 1 + 89.3T + 6.85e3T^{2} \)
23 \( 1 + (172. - 99.7i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-25.2 + 43.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 290. iT - 5.06e4T^{2} \)
41 \( 1 + (-26.6 - 46.1i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-258. - 149. i)T + (3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (362. + 209. i)T + (5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 399. iT - 1.48e5T^{2} \)
59 \( 1 + (-49.1 - 85.0i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-341. + 592. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (195. - 112. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 512.T + 3.57e5T^{2} \)
73 \( 1 + 994. iT - 3.89e5T^{2} \)
79 \( 1 + (100. - 174. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (959. + 553. i)T + (2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 372.T + 7.04e5T^{2} \)
97 \( 1 + (120. + 69.6i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.26848129670267832147822431452, −11.56120369530880409585832563278, −10.30842021647982782145817935460, −9.576300650747799816321712598392, −7.959589918452634815317525623170, −6.79162021203449952306102311711, −6.02700524145191875160344736539, −4.97479886292797345066595949024, −4.18844254220483479293381316940, −1.81692273881956616180767485236, 0.083468554658726522300315065908, 2.38729560536778823213696214316, 4.04036953688974081777502482038, 4.81940249351946668265277639299, 5.79246697034304836259284980008, 7.21169537150014652247388304901, 8.297296014333885858265113167123, 9.746639643386980213857909286644, 10.64122381584368440869744228798, 11.51798661857571405043099431325

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