Properties

Label 2-15e2-45.34-c3-0-29
Degree $2$
Conductor $225$
Sign $0.340 + 0.940i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.340 + 0.940i$
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: \((2,\ 225,\ (\ :3/2),\ 0.340 + 0.940i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26132 - 0.885199i\)
\(L(\frac12)\) \(\approx\) \(1.26132 - 0.885199i\)
\(L(\frac{5}{2})\) 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

−11.04145049801441368825001822998, −10.59259418126264028475849060579, −9.582673649764419290528289499941, −8.726679397480980307813868194319, −8.157851967964785475365819741875, −6.67166795023279750783834074437, −5.18670299184194647832371715006, −3.75804069887898372324694383078, −2.43721871799943055814180465258, −0.850467030198802479377128230569, 1.36409920659621904443395869350, 3.20268148057777608424245311848, 4.22021399891771980943251916758, 6.34909513224692296361951102896, 7.09316767380020406858133181471, 8.295008401763010173756195970081, 8.838867089406748887521849226158, 9.570726829115728981022204764535, 10.79806456492244600787678087669, 12.17629726928603040959073256646

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