# Properties

 Label 2-136-136.123-c0-0-0 Degree $2$ Conductor $136$ Sign $-0.615 - 0.788i$ Analytic cond. $0.0678728$ Root an. cond. $0.260524$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + i·2-s + (−1 + i)3-s − 4-s + (−1 − i)6-s − i·8-s − i·9-s + (1 + i)11-s + (1 − i)12-s + 16-s + i·17-s + 18-s − 2i·19-s + (−1 + i)22-s + (1 + i)24-s − i·25-s + ⋯
 L(s)  = 1 + i·2-s + (−1 + i)3-s − 4-s + (−1 − i)6-s − i·8-s − i·9-s + (1 + i)11-s + (1 − i)12-s + 16-s + i·17-s + 18-s − 2i·19-s + (−1 + i)22-s + (1 + i)24-s − i·25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$136$$    =    $$2^{3} \cdot 17$$ Sign: $-0.615 - 0.788i$ Analytic conductor: $$0.0678728$$ Root analytic conductor: $$0.260524$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{136} (123, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 136,\ (\ :0),\ -0.615 - 0.788i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.4499568977$$ $$L(\frac12)$$ $$\approx$$ $$0.4499568977$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - iT$$
17 $$1 - iT$$
good3 $$1 + (1 - i)T - iT^{2}$$
5 $$1 + iT^{2}$$
7 $$1 - iT^{2}$$
11 $$1 + (-1 - i)T + iT^{2}$$
13 $$1 - T^{2}$$
19 $$1 + 2iT - T^{2}$$
23 $$1 - iT^{2}$$
29 $$1 + iT^{2}$$
31 $$1 + iT^{2}$$
37 $$1 + iT^{2}$$
41 $$1 + (1 + i)T + iT^{2}$$
43 $$1 - T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - iT^{2}$$
67 $$1 + T^{2}$$
71 $$1 + iT^{2}$$
73 $$1 + (1 - i)T - iT^{2}$$
79 $$1 - iT^{2}$$
83 $$1 - T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + (-1 + i)T - 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

−14.10829379018122111854593772233, −12.83260234817688541120987881577, −11.77449855381196807410731862227, −10.57382924601604411035752812235, −9.663511030670705522782666378170, −8.718488301401035354803204082389, −7.09243364214847237611653603262, −6.15309186342952232387332769817, −4.89202963436691088873700380710, −4.12555615364533978441904967912, 1.42085106340463447177544392325, 3.55644053736731572668032407830, 5.33625654831850659557344740679, 6.35973643123559432573863254683, 7.80649168776885473376423669888, 9.083110844705150311940738088582, 10.31820460079360129722653774897, 11.58875341989955402464879387225, 11.76561960022422668253953730470, 12.85715835520574200022236771097