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Properties

Label 2-15e2-9.4-c3-0-15
Degree $2$
Conductor $225$
Sign $0.862 - 0.506i$
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  + (−0.669 − 1.16i)2-s + (4.26 + 2.96i)3-s + (3.10 − 5.37i)4-s + (0.580 − 6.93i)6-s + (11.1 + 19.3i)7-s − 19.0·8-s + (9.42 + 25.3i)9-s + (14.3 + 24.7i)11-s + (29.1 − 13.7i)12-s + (−37.8 + 65.5i)13-s + (14.9 − 25.9i)14-s + (−12.0 − 20.9i)16-s + 126.·17-s + (23.0 − 27.8i)18-s − 72.6·19-s + ⋯
L(s)  = 1  + (−0.236 − 0.410i)2-s + (0.821 + 0.570i)3-s + (0.387 − 0.671i)4-s + (0.0394 − 0.471i)6-s + (0.602 + 1.04i)7-s − 0.841·8-s + (0.349 + 0.937i)9-s + (0.392 + 0.679i)11-s + (0.701 − 0.330i)12-s + (−0.807 + 1.39i)13-s + (0.285 − 0.494i)14-s + (−0.188 − 0.326i)16-s + 1.80·17-s + (0.301 − 0.365i)18-s − 0.877·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.862 - 0.506i$
Analytic conductor: \(13.2754\)
Root analytic conductor: \(3.64354\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :3/2),\ 0.862 - 0.506i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.24152 + 0.609726i\)
\(L(\frac12)\) \(\approx\) \(2.24152 + 0.609726i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.26 - 2.96i)T \)
5 \( 1 \)
good2 \( 1 + (0.669 + 1.16i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-11.1 - 19.3i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-14.3 - 24.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (37.8 - 65.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 126.T + 4.91e3T^{2} \)
19 \( 1 + 72.6T + 6.85e3T^{2} \)
23 \( 1 + (-62.0 + 107. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (41.8 + 72.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (18.9 - 32.7i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 256.T + 5.06e4T^{2} \)
41 \( 1 + (-42.3 + 73.3i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-16.4 - 28.5i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-31.2 - 54.0i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 310.T + 1.48e5T^{2} \)
59 \( 1 + (-100. + 173. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (246. + 427. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-93.8 + 162. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.00e3T + 3.57e5T^{2} \)
73 \( 1 + 930.T + 3.89e5T^{2} \)
79 \( 1 + (-410. - 711. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (416. + 721. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 537.T + 7.04e5T^{2} \)
97 \( 1 + (-53.1 - 92.0i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.81960447261748755160695719394, −10.80166128898043767741839898801, −9.723617102444354577767589175619, −9.271226606022055124743418655490, −8.187409295839414991511917997720, −6.87746051214982520436596674199, −5.46991377633578480862146163045, −4.38061233963857652317565375657, −2.62808218728003097379730869052, −1.77193026075852594617193498972, 1.01017791189134615379615359716, 2.86784479644019950349821033125, 3.82046257339647311551695603437, 5.72662173918291007502558528773, 7.12218320775183371226668222024, 7.72716533856025368528574957745, 8.318291131002918871732154321612, 9.570963999328879101628710028876, 10.73260911813905359654315414093, 11.89671879627500272751417236609

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