Properties

 Degree $2$ Conductor $1225$ Sign $0.496 - 0.867i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.67 + 0.448i)2-s + (1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (−0.866 − 0.5i)9-s + (0.5 + 0.866i)11-s + (0.500 − 0.866i)16-s + (1.67 + 0.448i)18-s + (−1.22 − 1.22i)22-s + (0.448 + 1.67i)23-s + i·29-s − 2·36-s + (1.67 − 0.448i)37-s + (1.22 − 1.22i)43-s + (1.73 + 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯
 L(s)  = 1 + (−1.67 + 0.448i)2-s + (1.73 − 1.00i)4-s + (−1.22 + 1.22i)8-s + (−0.866 − 0.5i)9-s + (0.5 + 0.866i)11-s + (0.500 − 0.866i)16-s + (1.67 + 0.448i)18-s + (−1.22 − 1.22i)22-s + (0.448 + 1.67i)23-s + i·29-s − 2·36-s + (1.67 − 0.448i)37-s + (1.22 − 1.22i)43-s + (1.73 + 0.999i)44-s + (−1.50 − 2.59i)46-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1225$$    =    $$5^{2} \cdot 7^{2}$$ Sign: $0.496 - 0.867i$ Motivic weight: $$0$$ Character: $\chi_{1225} (1157, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1225,\ (\ :0),\ 0.496 - 0.867i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4493610666$$ $$L(\frac12)$$ $$\approx$$ $$0.4493610666$$ $$L(1)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1$$
7 $$1$$
good2 $$1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2}$$
3 $$1 + (0.866 + 0.5i)T^{2}$$
11 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + (-0.866 - 0.5i)T^{2}$$
19 $$1 + (0.5 + 0.866i)T^{2}$$
23 $$1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2}$$
29 $$1 - iT - T^{2}$$
31 $$1 + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2}$$
41 $$1 + T^{2}$$
43 $$1 + (-1.22 + 1.22i)T - iT^{2}$$
47 $$1 + (0.866 - 0.5i)T^{2}$$
53 $$1 + (0.866 + 0.5i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 + (0.866 + 0.5i)T^{2}$$
79 $$1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 - iT^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + iT^{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

−9.642920884418739934945132279550, −9.278278982576833044602296419632, −8.591689426133857874931771599756, −7.63137124690433274908751214152, −7.08478272532707150749643069449, −6.20158632784032086414084765505, −5.35309891183750941152610222615, −3.84273784607651024876426960417, −2.46475113049638324653093542350, −1.19578509012443993254680863876, 0.795073279066333925922302132925, 2.31278192804378191283822101731, 3.06351547788081039922688455579, 4.53426284451375565156702819608, 5.95275891675322451766954676615, 6.63312760434713522313148700277, 7.87476645177958270627845876133, 8.207813644904337253833370774792, 9.045220977357553702149470134919, 9.602043827071134545731981088649