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Properties

Label 2-15e2-25.16-c1-0-3
Degree $2$
Conductor $225$
Sign $0.811 + 0.583i$
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  + (−0.757 − 0.550i)2-s + (−0.346 − 1.06i)4-s + (1.42 + 1.71i)5-s + 2.74·7-s + (−0.903 + 2.78i)8-s + (−0.136 − 2.09i)10-s + (−0.236 − 0.171i)11-s + (2.87 − 2.09i)13-s + (−2.07 − 1.50i)14-s + (0.401 − 0.291i)16-s + (1.35 − 4.16i)17-s + (0.789 − 2.43i)19-s + (1.33 − 2.12i)20-s + (0.0845 + 0.260i)22-s + (4.15 + 3.01i)23-s + ⋯
L(s)  = 1  + (−0.535 − 0.389i)2-s + (−0.173 − 0.533i)4-s + (0.639 + 0.768i)5-s + 1.03·7-s + (−0.319 + 0.983i)8-s + (−0.0432 − 0.661i)10-s + (−0.0712 − 0.0517i)11-s + (0.797 − 0.579i)13-s + (−0.555 − 0.403i)14-s + (0.100 − 0.0728i)16-s + (0.327 − 1.00i)17-s + (0.181 − 0.557i)19-s + (0.299 − 0.474i)20-s + (0.0180 + 0.0554i)22-s + (0.865 + 0.628i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.811 + 0.583i$
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: \((2,\ 225,\ (\ :1/2),\ 0.811 + 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02977 - 0.331744i\)
\(L(\frac12)\) \(\approx\) \(1.02977 - 0.331744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.42 - 1.71i)T \)
good2 \( 1 + (0.757 + 0.550i)T + (0.618 + 1.90i)T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
11 \( 1 + (0.236 + 0.171i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-2.87 + 2.09i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.35 + 4.16i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.789 + 2.43i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-4.15 - 3.01i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.90 + 8.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.76 - 8.50i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.594 + 0.432i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.13 - 6.63i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.249T + 43T^{2} \)
47 \( 1 + (0.891 + 2.74i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.50 - 4.63i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.12 - 2.99i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-1.61 - 1.17i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (4.23 - 13.0i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (3.83 + 11.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (3.68 + 2.67i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.33 - 7.17i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.44 - 10.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (6.46 + 4.69i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.10 + 15.7i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.61514247024498242709716129310, −11.11302726389940750944961408875, −10.26584439454732500367976315068, −9.392503386386963368114582075446, −8.401357534630903744116172950443, −7.20756567722339728167602007194, −5.82836259426300769070019069014, −4.98183646669463453402131935734, −2.93885010622779813412587202447, −1.45906839797871157464102081116, 1.59453504782570337812342275029, 3.79635533237852141253720277807, 5.04153635899563886475465066315, 6.29375729386520509028657690202, 7.60016323004522013669583370129, 8.536079641756176184817159439548, 9.033173358803270490385142839490, 10.24373295746392760815930754167, 11.37673673241525249689949606407, 12.49577687342128827507203412520

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