# Properties

 Label 2-15e2-5.3-c2-0-9 Degree $2$ Conductor $225$ Sign $0.899 + 0.437i$ Analytic cond. $6.13080$ Root an. cond. $2.47604$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (1.22 − 1.22i)2-s + 1.00i·4-s + (4.89 − 4.89i)7-s + (6.12 + 6.12i)8-s + 3·11-s + (7.34 + 7.34i)13-s − 11.9i·14-s + 10.9·16-s + (13.4 − 13.4i)17-s − 5i·19-s + (3.67 − 3.67i)22-s + (17.1 + 17.1i)23-s + 18·26-s + (4.89 + 4.89i)28-s − 30i·29-s + ⋯
 L(s)  = 1 + (0.612 − 0.612i)2-s + 0.250i·4-s + (0.699 − 0.699i)7-s + (0.765 + 0.765i)8-s + 0.272·11-s + (0.565 + 0.565i)13-s − 0.857i·14-s + 0.687·16-s + (0.792 − 0.792i)17-s − 0.263i·19-s + (0.167 − 0.167i)22-s + (0.745 + 0.745i)23-s + 0.692·26-s + (0.174 + 0.174i)28-s − 1.03i·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$225$$    =    $$3^{2} \cdot 5^{2}$$ Sign: $0.899 + 0.437i$ Analytic conductor: $$6.13080$$ Root analytic conductor: $$2.47604$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{225} (118, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 225,\ (\ :1),\ 0.899 + 0.437i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.34347 - 0.539934i$$ $$L(\frac12)$$ $$\approx$$ $$2.34347 - 0.539934i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
5 $$1$$
good2 $$1 + (-1.22 + 1.22i)T - 4iT^{2}$$
7 $$1 + (-4.89 + 4.89i)T - 49iT^{2}$$
11 $$1 - 3T + 121T^{2}$$
13 $$1 + (-7.34 - 7.34i)T + 169iT^{2}$$
17 $$1 + (-13.4 + 13.4i)T - 289iT^{2}$$
19 $$1 + 5iT - 361T^{2}$$
23 $$1 + (-17.1 - 17.1i)T + 529iT^{2}$$
29 $$1 + 30iT - 841T^{2}$$
31 $$1 + 38T + 961T^{2}$$
37 $$1 + (19.5 - 19.5i)T - 1.36e3iT^{2}$$
41 $$1 + 57T + 1.68e3T^{2}$$
43 $$1 + (4.89 + 4.89i)T + 1.84e3iT^{2}$$
47 $$1 + (-7.34 + 7.34i)T - 2.20e3iT^{2}$$
53 $$1 + (31.8 + 31.8i)T + 2.80e3iT^{2}$$
59 $$1 - 90iT - 3.48e3T^{2}$$
61 $$1 + 28T + 3.72e3T^{2}$$
67 $$1 + (-47.7 + 47.7i)T - 4.48e3iT^{2}$$
71 $$1 + 42T + 5.04e3T^{2}$$
73 $$1 + (-13.4 - 13.4i)T + 5.32e3iT^{2}$$
79 $$1 - 80iT - 6.24e3T^{2}$$
83 $$1 + (111. + 111. i)T + 6.88e3iT^{2}$$
89 $$1 + 15iT - 7.92e3T^{2}$$
97 $$1 + (-53.8 + 53.8i)T - 9.40e3iT^{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

−11.72290190959703315006143495457, −11.37242253197311315363200387928, −10.30913828669451554409643666451, −9.018918614642254109912797955193, −7.87686297864578694477561461757, −7.02211502905865862740468964520, −5.32998019900926768765505391095, −4.29146790071985406439510430553, −3.24975858714790882141130346495, −1.57094550524415515468253757216, 1.53185053427783047779234247959, 3.58728750946998076273035988448, 5.01976207111317051179412876354, 5.72655264966527574820639343107, 6.80711956296735100158460333586, 8.018394320683349624310175932110, 8.993812806430919426708364828428, 10.30585750009338147021470993588, 11.06242637237477068679528908373, 12.34736086186204518856581931638