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Properties

Label 2-15e2-9.5-c2-0-2
Degree $2$
Conductor $225$
Sign $-0.888 + 0.458i$
Analytic cond. $6.13080$
Root an. cond. $2.47604$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.600 + 0.346i)2-s + (−2.96 + 0.450i)3-s + (−1.75 + 3.04i)4-s + (1.62 − 1.29i)6-s + (6.14 + 10.6i)7-s − 5.21i·8-s + (8.59 − 2.67i)9-s + (−11.5 + 6.67i)11-s + (3.84 − 9.83i)12-s + (−0.865 + 1.49i)13-s + (−7.37 − 4.25i)14-s + (−5.23 − 9.06i)16-s − 5.84i·17-s + (−4.23 + 4.58i)18-s − 16.8·19-s + ⋯
L(s)  = 1  + (−0.300 + 0.173i)2-s + (−0.988 + 0.150i)3-s + (−0.439 + 0.762i)4-s + (0.270 − 0.216i)6-s + (0.877 + 1.51i)7-s − 0.651i·8-s + (0.954 − 0.296i)9-s + (−1.05 + 0.606i)11-s + (0.320 − 0.819i)12-s + (−0.0665 + 0.115i)13-s + (−0.526 − 0.303i)14-s + (−0.327 − 0.566i)16-s − 0.343i·17-s + (−0.235 + 0.254i)18-s − 0.884·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.888 + 0.458i$
Analytic conductor: \(6.13080\)
Root analytic conductor: \(2.47604\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1),\ -0.888 + 0.458i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0797150 - 0.328674i\)
\(L(\frac12)\) \(\approx\) \(0.0797150 - 0.328674i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.96 - 0.450i)T \)
5 \( 1 \)
good2 \( 1 + (0.600 - 0.346i)T + (2 - 3.46i)T^{2} \)
7 \( 1 + (-6.14 - 10.6i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (11.5 - 6.67i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (0.865 - 1.49i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 5.84iT - 289T^{2} \)
19 \( 1 + 16.8T + 361T^{2} \)
23 \( 1 + (28.8 + 16.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-28.4 + 16.4i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.0240 - 0.0415i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 29.7T + 1.36e3T^{2} \)
41 \( 1 + (-8.34 - 4.81i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-8.45 - 14.6i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-6.30 + 3.63i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 29.6iT - 2.80e3T^{2} \)
59 \( 1 + (79.7 + 46.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-10.0 - 17.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (17.4 - 30.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 83.7iT - 5.04e3T^{2} \)
73 \( 1 - 7.39T + 5.32e3T^{2} \)
79 \( 1 + (10.6 + 18.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-51.3 + 29.6i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 46.7iT - 7.92e3T^{2} \)
97 \( 1 + (-11.5 - 20.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.25069384224467457565714938686, −11.91461594397887014262416869107, −10.65836979525131461492692997533, −9.640431376218819702675438971352, −8.529356877152054547311465591321, −7.77485573679736972457140785616, −6.41785671396737322360727536297, −5.21104697453287018624491526954, −4.40024001062309845278899760478, −2.35509462201289096432093899472, 0.23573452262855791623659071801, 1.56481359308108160595168806935, 4.20620558929890384048634324478, 5.11787582062138092234313044151, 6.15836480187621837347925363994, 7.48438639987897608100345143086, 8.371434535830880684967700984340, 9.983238902388312415399102852915, 10.69572382132233782833143667647, 10.95710986857372677483871464706

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