# Properties

 Label 2-136-17.16-c3-0-7 Degree $2$ Conductor $136$ Sign $0.782 + 0.622i$ Analytic cond. $8.02425$ Root an. cond. $2.83271$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2.62i·3-s + 11.5i·5-s − 23.0i·7-s + 20.1·9-s − 1.19i·11-s + 58.3·13-s + 30.2·15-s + (−54.8 − 43.6i)17-s + 138.·19-s − 60.4·21-s − 180. i·23-s − 7.98·25-s − 123. i·27-s − 115. i·29-s + 210. i·31-s + ⋯
 L(s)  = 1 − 0.504i·3-s + 1.03i·5-s − 1.24i·7-s + 0.745·9-s − 0.0327i·11-s + 1.24·13-s + 0.520·15-s + (−0.782 − 0.622i)17-s + 1.67·19-s − 0.627·21-s − 1.64i·23-s − 0.0638·25-s − 0.880i·27-s − 0.736i·29-s + 1.22i·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$136$$    =    $$2^{3} \cdot 17$$ Sign: $0.782 + 0.622i$ Analytic conductor: $$8.02425$$ Root analytic conductor: $$2.83271$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{136} (33, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 136,\ (\ :3/2),\ 0.782 + 0.622i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.68076 - 0.586758i$$ $$L(\frac12)$$ $$\approx$$ $$1.68076 - 0.586758i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
17 $$1 + (54.8 + 43.6i)T$$
good3 $$1 + 2.62iT - 27T^{2}$$
5 $$1 - 11.5iT - 125T^{2}$$
7 $$1 + 23.0iT - 343T^{2}$$
11 $$1 + 1.19iT - 1.33e3T^{2}$$
13 $$1 - 58.3T + 2.19e3T^{2}$$
19 $$1 - 138.T + 6.85e3T^{2}$$
23 $$1 + 180. iT - 1.21e4T^{2}$$
29 $$1 + 115. iT - 2.43e4T^{2}$$
31 $$1 - 210. iT - 2.97e4T^{2}$$
37 $$1 - 210. iT - 5.06e4T^{2}$$
41 $$1 - 297. iT - 6.89e4T^{2}$$
43 $$1 - 174.T + 7.95e4T^{2}$$
47 $$1 + 199.T + 1.03e5T^{2}$$
53 $$1 + 706.T + 1.48e5T^{2}$$
59 $$1 + 182.T + 2.05e5T^{2}$$
61 $$1 + 489. iT - 2.26e5T^{2}$$
67 $$1 + 167.T + 3.00e5T^{2}$$
71 $$1 - 818. iT - 3.57e5T^{2}$$
73 $$1 - 1.09e3iT - 3.89e5T^{2}$$
79 $$1 - 110. iT - 4.93e5T^{2}$$
83 $$1 - 118.T + 5.71e5T^{2}$$
89 $$1 - 400.T + 7.04e5T^{2}$$
97 $$1 + 924. iT - 9.12e5T^{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

−12.82988161074864512167161257058, −11.42114461196812305156892536426, −10.66869468150050434325391888166, −9.754773591310759185388130246400, −8.121738153984586735638281539542, −7.02608184319071590435899080800, −6.50475756465951423525505010232, −4.47876022020932137234349562861, −3.09807130355150348641933600340, −1.10222334703488268174480279219, 1.52255253395124227206334714495, 3.63545946294933430942593375005, 5.00895300638379963717581892643, 5.94148685814517768947610144992, 7.68294401288215629448384176458, 9.030054106137236379466249608401, 9.339955632610303486266926163705, 10.86431362976493171304382198687, 11.90412383666398273629872678022, 12.83809028253263988130793244945