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Properties

Label 2-15e2-9.4-c3-0-7
Degree $2$
Conductor $225$
Sign $-0.444 - 0.895i$
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  + (1.48 + 2.57i)2-s + (−5.14 + 0.741i)3-s + (−0.430 + 0.744i)4-s + (−9.56 − 12.1i)6-s + (−12.4 − 21.6i)7-s + 21.2·8-s + (25.8 − 7.62i)9-s + (31.6 + 54.7i)11-s + (1.65 − 4.14i)12-s + (−45.2 + 78.3i)13-s + (37.1 − 64.3i)14-s + (35.0 + 60.7i)16-s + 10.2·17-s + (58.2 + 55.4i)18-s + 24.7·19-s + ⋯
L(s)  = 1  + (0.526 + 0.911i)2-s + (−0.989 + 0.142i)3-s + (−0.0537 + 0.0931i)4-s + (−0.650 − 0.826i)6-s + (−0.673 − 1.16i)7-s + 0.939·8-s + (0.959 − 0.282i)9-s + (0.866 + 1.50i)11-s + (0.0399 − 0.0998i)12-s + (−0.965 + 1.67i)13-s + (0.709 − 1.22i)14-s + (0.547 + 0.949i)16-s + 0.145·17-s + (0.762 + 0.725i)18-s + 0.298·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $-0.444 - 0.895i$
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: \((2,\ 225,\ (\ :3/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.834770 + 1.34668i\)
\(L(\frac12)\) \(\approx\) \(0.834770 + 1.34668i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.14 - 0.741i)T \)
5 \( 1 \)
good2 \( 1 + (-1.48 - 2.57i)T + (-4 + 6.92i)T^{2} \)
7 \( 1 + (12.4 + 21.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-31.6 - 54.7i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (45.2 - 78.3i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 10.2T + 4.91e3T^{2} \)
19 \( 1 - 24.7T + 6.85e3T^{2} \)
23 \( 1 + (1.34 - 2.33i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-54.8 - 94.9i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-34.7 + 60.1i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 23.0T + 5.06e4T^{2} \)
41 \( 1 + (133. - 230. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-235. - 407. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-218. - 378. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 114.T + 1.48e5T^{2} \)
59 \( 1 + (-237. + 410. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-166. - 288. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (38.0 - 65.9i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 658.T + 3.57e5T^{2} \)
73 \( 1 + 549.T + 3.89e5T^{2} \)
79 \( 1 + (-85.7 - 148. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (166. + 288. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 621.T + 7.04e5T^{2} \)
97 \( 1 + (759. + 1.31e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.18455506737783100985612942433, −11.22774664993237160285244282772, −10.01592528339507635139340218214, −9.588520515650932850657865780763, −7.23989392057577695536425399680, −7.05049474448880345175535399752, −6.15427252712275165094861075105, −4.59788132935231607685218332398, −4.28576157478462484167505611539, −1.44247369809461124035268813855, 0.67893988643421393938039725294, 2.55658125223597667682996082516, 3.64928007629210305457891422876, 5.27214032203521241913461556878, 5.94577845464672893643996485114, 7.29189098482683985051223487849, 8.617758612353637624505167813586, 9.995604213204669993414275972728, 10.74141924253257511453817237567, 11.87437072720429880776883051091

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