# Properties

 Label 2-75-15.14-c2-0-3 Degree $2$ Conductor $75$ Sign $0.262 + 0.964i$ Analytic cond. $2.04360$ Root an. cond. $1.42954$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.23·2-s + (−2.23 + 2i)3-s + 1.00·4-s + (5.00 − 4.47i)6-s − 6i·7-s + 6.70·8-s + (1.00 − 8.94i)9-s − 4.47i·11-s + (−2.23 + 2.00i)12-s − 16i·13-s + 13.4i·14-s − 19·16-s + 4.47·17-s + (−2.23 + 20.0i)18-s + 2·19-s + ⋯
 L(s)  = 1 − 1.11·2-s + (−0.745 + 0.666i)3-s + 0.250·4-s + (0.833 − 0.745i)6-s − 0.857i·7-s + 0.838·8-s + (0.111 − 0.993i)9-s − 0.406i·11-s + (−0.186 + 0.166i)12-s − 1.23i·13-s + 0.958i·14-s − 1.18·16-s + 0.263·17-s + (−0.124 + 1.11i)18-s + 0.105·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$75$$    =    $$3 \cdot 5^{2}$$ Sign: $0.262 + 0.964i$ Analytic conductor: $$2.04360$$ Root analytic conductor: $$1.42954$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{75} (74, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 75,\ (\ :1),\ 0.262 + 0.964i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.316987 - 0.242157i$$ $$L(\frac12)$$ $$\approx$$ $$0.316987 - 0.242157i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (2.23 - 2i)T$$
5 $$1$$
good2 $$1 + 2.23T + 4T^{2}$$
7 $$1 + 6iT - 49T^{2}$$
11 $$1 + 4.47iT - 121T^{2}$$
13 $$1 + 16iT - 169T^{2}$$
17 $$1 - 4.47T + 289T^{2}$$
19 $$1 - 2T + 361T^{2}$$
23 $$1 + 13.4T + 529T^{2}$$
29 $$1 + 31.3iT - 841T^{2}$$
31 $$1 + 18T + 961T^{2}$$
37 $$1 + 16iT - 1.36e3T^{2}$$
41 $$1 + 62.6iT - 1.68e3T^{2}$$
43 $$1 + 16iT - 1.84e3T^{2}$$
47 $$1 + 49.1T + 2.20e3T^{2}$$
53 $$1 + 4.47T + 2.80e3T^{2}$$
59 $$1 - 4.47iT - 3.48e3T^{2}$$
61 $$1 - 82T + 3.72e3T^{2}$$
67 $$1 - 24iT - 4.48e3T^{2}$$
71 $$1 - 125. iT - 5.04e3T^{2}$$
73 $$1 - 74iT - 5.32e3T^{2}$$
79 $$1 + 138T + 6.24e3T^{2}$$
83 $$1 - 93.9T + 6.88e3T^{2}$$
89 $$1 - 107. iT - 7.92e3T^{2}$$
97 $$1 + 166iT - 9.40e3T^{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

−14.13606309521461343941098688070, −12.88900305660941201977522951769, −11.37453480849247090349298636590, −10.42312508423581756441210393947, −9.835576344788790113193440943264, −8.465249663615244286862819829368, −7.24552569594067322114938871160, −5.55406273432819310902086409115, −3.96647598190035573314713142173, −0.56232865399743246886736453768, 1.76146658001021838191090524086, 4.87465527927019943797374885879, 6.46956883220554981972872824927, 7.66099038606770347015452052937, 8.840780556043446652023125369066, 9.918549321675346939345970827289, 11.19681487127270218773171017456, 12.10494377070694800026312630685, 13.25256448154331197846827024961, 14.49232144727837867357274384904