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Properties

Label 2-15e2-25.11-c1-0-10
Degree $2$
Conductor $225$
Sign $0.0252 + 0.999i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.74 − 1.26i)2-s + (0.824 − 2.53i)4-s + (0.460 − 2.18i)5-s − 3.16·7-s + (−0.446 − 1.37i)8-s + (−1.97 − 4.40i)10-s + (1.24 − 0.904i)11-s + (4.24 + 3.08i)13-s + (−5.52 + 4.01i)14-s + (1.79 + 1.30i)16-s + (−0.398 − 1.22i)17-s + (1.68 + 5.17i)19-s + (−5.17 − 2.97i)20-s + (1.02 − 3.16i)22-s + (−5.21 + 3.78i)23-s + ⋯
L(s)  = 1  + (1.23 − 0.897i)2-s + (0.412 − 1.26i)4-s + (0.205 − 0.978i)5-s − 1.19·7-s + (−0.157 − 0.485i)8-s + (−0.624 − 1.39i)10-s + (0.375 − 0.272i)11-s + (1.17 + 0.854i)13-s + (−1.47 + 1.07i)14-s + (0.448 + 0.325i)16-s + (−0.0967 − 0.297i)17-s + (0.386 + 1.18i)19-s + (−1.15 − 0.664i)20-s + (0.218 − 0.673i)22-s + (−1.08 + 0.789i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0252 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0252 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.0252 + 0.999i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.0252 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54392 - 1.50537i\)
\(L(\frac12)\) \(\approx\) \(1.54392 - 1.50537i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.460 + 2.18i)T \)
good2 \( 1 + (-1.74 + 1.26i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 + (-1.24 + 0.904i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.24 - 3.08i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.398 + 1.22i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.68 - 5.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (5.21 - 3.78i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.730 + 2.24i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.37 + 4.24i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.81 - 3.50i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.90 + 5.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + (0.232 - 0.716i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.01 - 9.27i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.32 - 2.41i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-8.65 + 6.28i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.586 + 1.80i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-0.0219 + 0.0674i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.24 - 2.35i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.500 + 1.53i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.06 + 12.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-5.88 + 4.27i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.15 + 9.69i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.99964225735291804829180744470, −11.59005591762431583195003656277, −10.16730491300193395110768252675, −9.378527114746304610402383936404, −8.193362825982061873020197244348, −6.32555324560957680432337726758, −5.64053426945079596611971251018, −4.20142101103515936543659870176, −3.45387970012373376762701208681, −1.67378115233848362894739534144, 3.01541187740288564806278834308, 3.85356262871722661522661470272, 5.42079392919973021409711416691, 6.52367564301245560406301019767, 6.76235372937483398378656735018, 8.191594803555265705938266139046, 9.688075296119372006169544114011, 10.57594959069173356195744959548, 11.80183938672957985102447574835, 13.00765853109400874587153201456

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