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Properties

Label 2-15e2-25.16-c1-0-9
Degree $2$
Conductor $225$
Sign $0.728 + 0.684i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.363i)2-s + (−0.5 − 1.53i)4-s + (1.80 − 1.31i)5-s − 1.61·7-s + (0.690 − 2.12i)8-s + 1.38·10-s + (−0.618 − 0.449i)11-s + (3.92 − 2.85i)13-s + (−0.809 − 0.587i)14-s + (−1.49 + 1.08i)16-s + (0.236 − 0.726i)17-s + (−1.80 + 5.56i)19-s + (−2.92 − 2.12i)20-s + (−0.145 − 0.449i)22-s + (6.66 + 4.84i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.256i)2-s + (−0.250 − 0.769i)4-s + (0.809 − 0.587i)5-s − 0.611·7-s + (0.244 − 0.751i)8-s + 0.437·10-s + (−0.186 − 0.135i)11-s + (1.08 − 0.791i)13-s + (−0.216 − 0.157i)14-s + (−0.374 + 0.272i)16-s + (0.0572 − 0.176i)17-s + (−0.415 + 1.27i)19-s + (−0.654 − 0.475i)20-s + (−0.0311 − 0.0957i)22-s + (1.38 + 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :1/2),\ 0.728 + 0.684i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37414 - 0.544063i\)
\(L(\frac12)\) \(\approx\) \(1.37414 - 0.544063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.80 + 1.31i)T \)
good2 \( 1 + (-0.5 - 0.363i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + (0.618 + 0.449i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.236 + 0.726i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.66 - 4.84i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.427 - 1.31i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.927 - 2.85i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.42 - 2.48i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.23 - 3.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (1.69 + 5.20i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.35 - 2.43i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-3.80 - 2.76i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-2.85 + 8.78i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.35 - 4.16i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-7.28 - 5.29i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.954 - 2.93i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.545 + 1.67i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-7.23 - 5.25i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.881 - 2.71i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.53775755555082256544397166040, −10.94305985193043101298694795593, −10.11955146901339325881945974530, −9.327062545902336255436053825460, −8.319162286243845419043953168407, −6.68447540346646701823731038795, −5.80886779301057295495269907811, −5.06338717477443100141975868325, −3.47644320713367639394951751067, −1.32859984070497860860220619301, 2.38989452429082209185445301018, 3.52432800886435025671048143092, 4.86224161834378006994345301739, 6.32326455311551958490371033256, 7.12885805024586997477797125196, 8.646476486198490560046529505835, 9.323735275702216722958349624087, 10.67063963164402518415008856463, 11.32459953832312636825883890146, 12.60551758536625165547702597367

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