# Properties

 Label 2-234-117.11-c1-0-0 Degree $2$ Conductor $234$ Sign $0.704 - 0.709i$ Analytic cond. $1.86849$ Root an. cond. $1.36693$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.707 + 0.707i)2-s + (−1.69 − 0.356i)3-s − 1.00i·4-s + (−3.71 − 0.995i)5-s + (1.45 − 0.946i)6-s + (2.11 + 0.566i)7-s + (0.707 + 0.707i)8-s + (2.74 + 1.20i)9-s + (3.33 − 1.92i)10-s + (2.96 + 2.96i)11-s + (−0.356 + 1.69i)12-s + (3.17 + 1.70i)13-s + (−1.89 + 1.09i)14-s + (5.94 + 3.00i)15-s − 1.00·16-s + (0.465 − 0.806i)17-s + ⋯
 L(s)  = 1 + (−0.499 + 0.499i)2-s + (−0.978 − 0.205i)3-s − 0.500i·4-s + (−1.66 − 0.445i)5-s + (0.592 − 0.386i)6-s + (0.798 + 0.214i)7-s + (0.250 + 0.250i)8-s + (0.915 + 0.402i)9-s + (1.05 − 0.607i)10-s + (0.892 + 0.892i)11-s + (−0.102 + 0.489i)12-s + (0.880 + 0.473i)13-s + (−0.506 + 0.292i)14-s + (1.53 + 0.777i)15-s − 0.250·16-s + (0.112 − 0.195i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$234$$    =    $$2 \cdot 3^{2} \cdot 13$$ Sign: $0.704 - 0.709i$ Analytic conductor: $$1.86849$$ Root analytic conductor: $$1.36693$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{234} (11, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 234,\ (\ :1/2),\ 0.704 - 0.709i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.554161 + 0.230782i$$ $$L(\frac12)$$ $$\approx$$ $$0.554161 + 0.230782i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.707 - 0.707i)T$$
3 $$1 + (1.69 + 0.356i)T$$
13 $$1 + (-3.17 - 1.70i)T$$
good5 $$1 + (3.71 + 0.995i)T + (4.33 + 2.5i)T^{2}$$
7 $$1 + (-2.11 - 0.566i)T + (6.06 + 3.5i)T^{2}$$
11 $$1 + (-2.96 - 2.96i)T + 11iT^{2}$$
17 $$1 + (-0.465 + 0.806i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (1.86 - 0.501i)T + (16.4 - 9.5i)T^{2}$$
23 $$1 + (-3.91 + 6.78i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 - 7.40iT - 29T^{2}$$
31 $$1 + (0.333 - 1.24i)T + (-26.8 - 15.5i)T^{2}$$
37 $$1 + (-4.93 - 1.32i)T + (32.0 + 18.5i)T^{2}$$
41 $$1 + (-2.83 - 10.5i)T + (-35.5 + 20.5i)T^{2}$$
43 $$1 + (-1.70 + 0.984i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (-9.40 + 2.52i)T + (40.7 - 23.5i)T^{2}$$
53 $$1 - 2.04iT - 53T^{2}$$
59 $$1 + (4.46 + 4.46i)T + 59iT^{2}$$
61 $$1 + (0.430 + 0.745i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (3.23 - 0.865i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 + (-2.90 - 10.8i)T + (-61.4 + 35.5i)T^{2}$$
73 $$1 + (2.45 - 2.45i)T - 73iT^{2}$$
79 $$1 + (-6.38 + 11.0i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (1.45 + 5.43i)T + (-71.8 + 41.5i)T^{2}$$
89 $$1 + (-0.522 + 1.95i)T + (-77.0 - 44.5i)T^{2}$$
97 $$1 + (4.55 - 17.0i)T + (-84.0 - 48.5i)T^{2}$$
show less
$$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

−12.07578998148154646789788989661, −11.35448341683894467127814937497, −10.68656528702851839675689342785, −9.102706514636723040329597019705, −8.251313077538800319487731922614, −7.29074223056606425613505012454, −6.47173348881359645085253576011, −4.88176992001524119399789837493, −4.22733521714075668291283585591, −1.19951222237010550292316409826, 0.868719491280413783597944810222, 3.57281658370701463493349188941, 4.28127848474147422436442164770, 5.94718790309612117201941503289, 7.27687359549650393725888161413, 8.043232303912022482491985705902, 9.139481520783847895941535917405, 10.67524389357872015855876694510, 11.19680634781020834959460306793, 11.58360264022619616107063731330