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Properties

Label 2-15e2-225.16-c1-0-7
Degree $2$
Conductor $225$
Sign $-0.276 - 0.961i$
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  + (−1.92 + 0.858i)2-s + (1.36 + 1.06i)3-s + (1.64 − 1.82i)4-s + (0.896 − 2.04i)5-s + (−3.55 − 0.868i)6-s + (−2.48 + 4.31i)7-s + (−0.295 + 0.907i)8-s + (0.751 + 2.90i)9-s + (0.0298 + 4.71i)10-s + (1.72 − 0.768i)11-s + (4.17 − 0.756i)12-s + (3.32 + 1.48i)13-s + (1.09 − 10.4i)14-s + (3.39 − 1.85i)15-s + (0.302 + 2.87i)16-s + (−1.29 + 3.97i)17-s + ⋯
L(s)  = 1  + (−1.36 + 0.606i)2-s + (0.790 + 0.612i)3-s + (0.820 − 0.911i)4-s + (0.400 − 0.916i)5-s + (−1.44 − 0.354i)6-s + (−0.940 + 1.62i)7-s + (−0.104 + 0.321i)8-s + (0.250 + 0.968i)9-s + (0.00943 + 1.49i)10-s + (0.520 − 0.231i)11-s + (1.20 − 0.218i)12-s + (0.922 + 0.410i)13-s + (0.293 − 2.79i)14-s + (0.877 − 0.478i)15-s + (0.0755 + 0.718i)16-s + (−0.313 + 0.964i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.276 - 0.961i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ -0.276 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.475997 + 0.632060i\)
\(L(\frac12)\) \(\approx\) \(0.475997 + 0.632060i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.36 - 1.06i)T \)
5 \( 1 + (-0.896 + 2.04i)T \)
good2 \( 1 + (1.92 - 0.858i)T + (1.33 - 1.48i)T^{2} \)
7 \( 1 + (2.48 - 4.31i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.72 + 0.768i)T + (7.36 - 8.17i)T^{2} \)
13 \( 1 + (-3.32 - 1.48i)T + (8.69 + 9.66i)T^{2} \)
17 \( 1 + (1.29 - 3.97i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.0810 - 0.249i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.257 - 2.45i)T + (-22.4 - 4.78i)T^{2} \)
29 \( 1 + (1.68 + 0.357i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-1.27 + 0.269i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (-6.88 + 5.00i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.39 + 1.51i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (1.88 - 3.26i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.60 - 0.552i)T + (42.9 + 19.1i)T^{2} \)
53 \( 1 + (3.69 + 11.3i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-10.1 - 4.50i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (-5.65 + 2.51i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (-10.8 + 2.30i)T + (61.2 - 27.2i)T^{2} \)
71 \( 1 + (2.45 + 7.54i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-4.25 - 3.09i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.33 - 1.13i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (3.11 + 3.46i)T + (-8.67 + 82.5i)T^{2} \)
89 \( 1 + (-0.287 - 0.208i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (9.60 + 2.04i)T + (88.6 + 39.4i)T^{2} \)
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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

−12.67736547542824873212809872346, −11.27952062783281978872178556253, −9.872214826879891664431752168732, −9.370930742100567212119590091004, −8.679025169039481719736785991650, −8.239780128166655804543426510870, −6.49443395535235924507667697009, −5.62384961478618943013620819827, −3.81188398036700460769150085623, −1.94989435558671863850182478439, 1.00909949268218733664360597664, 2.67031884250570327129156186330, 3.74047246773990422444148552988, 6.53001639402663641376368542829, 7.11737624839059647871861834566, 8.055830172697432486653949169566, 9.281968556219836982156290378505, 9.922542370553319357568121551781, 10.67022828539248878499816781285, 11.60518864060764172943892688880

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