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Properties

Label 2-15e2-45.4-c3-0-15
Degree $2$
Conductor $225$
Sign $0.113 - 0.993i$
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  + (0.195 − 0.112i)2-s + (−4.76 + 2.06i)3-s + (−3.97 + 6.88i)4-s + (−0.697 + 0.940i)6-s + (27.0 − 15.5i)7-s + 3.59i·8-s + (18.4 − 19.7i)9-s + (−9.06 − 15.6i)11-s + (4.71 − 41.0i)12-s + (43.4 + 25.0i)13-s + (3.51 − 6.08i)14-s + (−31.3 − 54.3i)16-s + 131. i·17-s + (1.37 − 5.92i)18-s − 23.2·19-s + ⋯
L(s)  = 1  + (0.0689 − 0.0398i)2-s + (−0.917 + 0.397i)3-s + (−0.496 + 0.860i)4-s + (−0.0474 + 0.0639i)6-s + (1.45 − 0.842i)7-s + 0.158i·8-s + (0.683 − 0.730i)9-s + (−0.248 − 0.430i)11-s + (0.113 − 0.987i)12-s + (0.926 + 0.535i)13-s + (0.0670 − 0.116i)14-s + (−0.490 − 0.849i)16-s + 1.87i·17-s + (0.0180 − 0.0775i)18-s − 0.280·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.113 - 0.993i$
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: \((2,\ 225,\ (\ :3/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.967006 + 0.862463i\)
\(L(\frac12)\) \(\approx\) \(0.967006 + 0.862463i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.76 - 2.06i)T \)
5 \( 1 \)
good2 \( 1 + (-0.195 + 0.112i)T + (4 - 6.92i)T^{2} \)
7 \( 1 + (-27.0 + 15.5i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (9.06 + 15.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-43.4 - 25.0i)T + (1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 131. iT - 4.91e3T^{2} \)
19 \( 1 + 23.2T + 6.85e3T^{2} \)
23 \( 1 + (-28.5 - 16.4i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-62.9 - 108. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (62.5 - 108. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 99.9iT - 5.06e4T^{2} \)
41 \( 1 + (122. - 212. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-120. + 69.5i)T + (3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-409. + 236. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 421. iT - 1.48e5T^{2} \)
59 \( 1 + (371. - 642. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (4.48 + 7.77i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-510. - 294. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 48.5T + 3.57e5T^{2} \)
73 \( 1 - 409. iT - 3.89e5T^{2} \)
79 \( 1 + (-265. - 459. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (255. - 147. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 852.T + 7.04e5T^{2} \)
97 \( 1 + (-336. + 194. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−11.91954794185540660858506225741, −10.94591126778425063934239985473, −10.55651043340790827185241176927, −8.885232023074609074832461077326, −8.153673318723225459616870781812, −7.00159039982245476742946252985, −5.62491485424544397421402050278, −4.43022947892390594481761696625, −3.79215585467526443981267857930, −1.29848408443658790696023179262, 0.70711859478107532151503729046, 2.06019081545897980338826365704, 4.64402377376472723693588867738, 5.24077799284999384940504580433, 6.14237739723586669061279002900, 7.50903180177448337072278312534, 8.590606253146384596437514290568, 9.729899516012709357664162295320, 10.89169074126209429372592611454, 11.41614385191418488019273156404

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