# Properties

 Label 2-72-8.5-c7-0-5 Degree $2$ Conductor $72$ Sign $-0.0484 - 0.998i$ Analytic cond. $22.4917$ Root an. cond. $4.74254$ Motivic weight $7$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (9.70 − 5.81i)2-s + (60.3 − 112. i)4-s + 324. i·5-s − 956.·7-s + (−70.1 − 1.44e3i)8-s + (1.88e3 + 3.14e3i)10-s + 5.45e3i·11-s + 6.28e3i·13-s + (−9.28e3 + 5.56e3i)14-s + (−9.09e3 − 1.36e4i)16-s − 3.45e4·17-s + 1.45e4i·19-s + (3.66e4 + 1.95e4i)20-s + (3.17e4 + 5.29e4i)22-s + 2.46e4·23-s + ⋯
 L(s)  = 1 + (0.857 − 0.513i)2-s + (0.471 − 0.881i)4-s + 1.16i·5-s − 1.05·7-s + (−0.0484 − 0.998i)8-s + (0.596 + 0.995i)10-s + 1.23i·11-s + 0.793i·13-s + (−0.904 + 0.541i)14-s + (−0.554 − 0.831i)16-s − 1.70·17-s + 0.488i·19-s + (1.02 + 0.547i)20-s + (0.634 + 1.05i)22-s + 0.422·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0484 - 0.998i)\, \overline{\Lambda}(8-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0484 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$72$$    =    $$2^{3} \cdot 3^{2}$$ Sign: $-0.0484 - 0.998i$ Analytic conductor: $$22.4917$$ Root analytic conductor: $$4.74254$$ Motivic weight: $$7$$ Rational: no Arithmetic: yes Character: $\chi_{72} (37, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 72,\ (\ :7/2),\ -0.0484 - 0.998i)$$

## Particular Values

 $$L(4)$$ $$\approx$$ $$1.15702 + 1.21446i$$ $$L(\frac12)$$ $$\approx$$ $$1.15702 + 1.21446i$$ $$L(\frac{9}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-9.70 + 5.81i)T$$
3 $$1$$
good5 $$1 - 324. iT - 7.81e4T^{2}$$
7 $$1 + 956.T + 8.23e5T^{2}$$
11 $$1 - 5.45e3iT - 1.94e7T^{2}$$
13 $$1 - 6.28e3iT - 6.27e7T^{2}$$
17 $$1 + 3.45e4T + 4.10e8T^{2}$$
19 $$1 - 1.45e4iT - 8.93e8T^{2}$$
23 $$1 - 2.46e4T + 3.40e9T^{2}$$
29 $$1 - 1.71e5iT - 1.72e10T^{2}$$
31 $$1 - 1.11e5T + 2.75e10T^{2}$$
37 $$1 - 1.03e5iT - 9.49e10T^{2}$$
41 $$1 + 7.16e4T + 1.94e11T^{2}$$
43 $$1 - 3.28e5iT - 2.71e11T^{2}$$
47 $$1 + 1.19e5T + 5.06e11T^{2}$$
53 $$1 + 1.04e6iT - 1.17e12T^{2}$$
59 $$1 + 2.25e5iT - 2.48e12T^{2}$$
61 $$1 + 1.55e6iT - 3.14e12T^{2}$$
67 $$1 + 3.16e5iT - 6.06e12T^{2}$$
71 $$1 + 5.38e5T + 9.09e12T^{2}$$
73 $$1 + 2.68e6T + 1.10e13T^{2}$$
79 $$1 - 8.22e6T + 1.92e13T^{2}$$
83 $$1 + 5.89e6iT - 2.71e13T^{2}$$
89 $$1 + 4.37e5T + 4.42e13T^{2}$$
97 $$1 + 7.84e6T + 8.07e13T^{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

−13.38954785617075056366465466215, −12.48107137445936482129495947049, −11.29453106634439176504928487295, −10.33139509632270565955506449903, −9.368990626694628900530027430509, −6.91495388914397436267461098013, −6.51850787555933153418523545073, −4.60761908092862012056191107797, −3.24962731139650811800349523121, −2.05307519545334511973140157664, 0.40397294169148143363687917825, 2.82442743394406087709862844087, 4.27014392347832649548468165581, 5.57776960157929975579076387267, 6.63268587082342441404846244904, 8.249835525526135810178408816272, 9.131361520225358944869146114252, 10.94267238325725734511494669486, 12.18331659167327056104803139836, 13.28777633384057606783632631489