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Properties

Label 2-15e2-9.4-c3-0-35
Degree $2$
Conductor $225$
Sign $-0.820 - 0.571i$
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  + (−2.42 − 4.20i)2-s + (4.79 − 1.99i)3-s + (−7.76 + 13.4i)4-s + (−20.0 − 15.3i)6-s + (8.11 + 14.0i)7-s + 36.5·8-s + (19.0 − 19.1i)9-s + (−29.7 − 51.5i)11-s + (−10.4 + 80.0i)12-s + (13.5 − 23.4i)13-s + (39.3 − 68.2i)14-s + (−26.5 − 45.9i)16-s − 78.9·17-s + (−126. − 33.5i)18-s − 142.·19-s + ⋯
L(s)  = 1  + (−0.857 − 1.48i)2-s + (0.923 − 0.384i)3-s + (−0.970 + 1.68i)4-s + (−1.36 − 1.04i)6-s + (0.438 + 0.759i)7-s + 1.61·8-s + (0.704 − 0.709i)9-s + (−0.815 − 1.41i)11-s + (−0.250 + 1.92i)12-s + (0.288 − 0.500i)13-s + (0.751 − 1.30i)14-s + (−0.414 − 0.718i)16-s − 1.12·17-s + (−1.65 − 0.438i)18-s − 1.71·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.820 - 0.571i$
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: \((2,\ 225,\ (\ :3/2),\ -0.820 - 0.571i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.281734 + 0.898396i\)
\(L(\frac12)\) \(\approx\) \(0.281734 + 0.898396i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.79 + 1.99i)T \)
5 \( 1 \)
good2 \( 1 + (2.42 + 4.20i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (-8.11 - 14.0i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (29.7 + 51.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-13.5 + 23.4i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 78.9T + 4.91e3T^{2} \)
19 \( 1 + 142.T + 6.85e3T^{2} \)
23 \( 1 + (-42.8 + 74.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (113. + 195. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-75.3 + 130. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 234.T + 5.06e4T^{2} \)
41 \( 1 + (36.9 - 63.9i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-133. - 231. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-133. - 230. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 603.T + 1.48e5T^{2} \)
59 \( 1 + (254. - 440. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (39.1 + 67.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-244. + 422. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 73.2T + 3.57e5T^{2} \)
73 \( 1 + 115.T + 3.89e5T^{2} \)
79 \( 1 + (391. + 677. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (135. + 235. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 - 211.T + 7.04e5T^{2} \)
97 \( 1 + (-833. - 1.44e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.05391317044230406301646371173, −10.41187268167782449849524447099, −9.061914426181493136622735940108, −8.561667652998618181007123675445, −7.942921821296019193657503252965, −6.12545501427319263204410199993, −4.15630848796590584041130252145, −2.81703407247886075907347927039, −2.12057180890417972661154279424, −0.44294632581066993103611380186, 1.91795180402368999802200594611, 4.19642015968598757577254235414, 5.11631288032266871026426171865, 6.93310052170580197135587647337, 7.29084680636839840358470698842, 8.480389298607448080082135050025, 9.007589684300379583393752480582, 10.17640079450224714941599922269, 10.74757084809137872153469606382, 12.75656697691346308482158517236

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