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Properties

Label 2-15e2-9.7-c3-0-16
Degree $2$
Conductor $225$
Sign $-0.0933 - 0.995i$
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  + (1.33 − 2.31i)2-s + (0.694 + 5.14i)3-s + (0.422 + 0.730i)4-s + (12.8 + 5.27i)6-s + (−7.70 + 13.3i)7-s + 23.6·8-s + (−26.0 + 7.14i)9-s + (−22.2 + 38.5i)11-s + (−3.47 + 2.68i)12-s + (−14.0 − 24.2i)13-s + (20.6 + 35.7i)14-s + (28.2 − 48.9i)16-s − 92.6·17-s + (−18.2 + 69.8i)18-s + 49.5·19-s + ⋯
L(s)  = 1  + (0.472 − 0.819i)2-s + (0.133 + 0.991i)3-s + (0.0527 + 0.0913i)4-s + (0.874 + 0.359i)6-s + (−0.416 + 0.720i)7-s + 1.04·8-s + (−0.964 + 0.264i)9-s + (−0.610 + 1.05i)11-s + (−0.0835 + 0.0644i)12-s + (−0.298 − 0.517i)13-s + (0.393 + 0.681i)14-s + (0.441 − 0.765i)16-s − 1.32·17-s + (−0.239 + 0.915i)18-s + 0.598·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0933 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.0933 - 0.995i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ -0.0933 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.26979 + 1.39436i\)
\(L(\frac12)\) \(\approx\) \(1.26979 + 1.39436i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.694 - 5.14i)T \)
5 \( 1 \)
good2 \( 1 + (-1.33 + 2.31i)T + (-4 - 6.92i)T^{2} \)
7 \( 1 + (7.70 - 13.3i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (22.2 - 38.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (14.0 + 24.2i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 92.6T + 4.91e3T^{2} \)
19 \( 1 - 49.5T + 6.85e3T^{2} \)
23 \( 1 + (0.461 + 0.799i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (94.9 - 164. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-149. - 259. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 - 57.8T + 5.06e4T^{2} \)
41 \( 1 + (-143. - 249. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-0.295 + 0.512i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-299. + 518. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + (-96.5 - 167. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-283. + 490. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-177. - 307. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 320.T + 3.57e5T^{2} \)
73 \( 1 - 636.T + 3.89e5T^{2} \)
79 \( 1 + (-143. + 249. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (-142. + 246. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 331.T + 7.04e5T^{2} \)
97 \( 1 + (910. - 1.57e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.07969477404422798254130192355, −11.05420249729793653734102786715, −10.29851127484022668556451925401, −9.413262637937821321519662807808, −8.305556268708328431214209999387, −7.01299326369053351843360536413, −5.32663640951623751894324678194, −4.49681154482479933087875742882, −3.18563979128337548213595260804, −2.30228689278458288227759801876, 0.62553377691628271418506065398, 2.39969692283075710520939334090, 4.15817865454229793365372316782, 5.66233603502684793701148503877, 6.43558681421870074855596997663, 7.32404506556925174037828274812, 8.101672593118663879744113636782, 9.447866630667513055800130544119, 10.80478593362961416989732753607, 11.53252699613636866336867507138

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