# Properties

 Label 2-69-1.1-c5-0-2 Degree $2$ Conductor $69$ Sign $1$ Analytic cond. $11.0664$ Root an. cond. $3.32663$ Motivic weight $5$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 3.54·2-s + 9·3-s − 19.4·4-s − 92.1·5-s − 31.8·6-s − 8.98·7-s + 182.·8-s + 81·9-s + 326.·10-s + 612.·11-s − 175.·12-s − 496.·13-s + 31.8·14-s − 829.·15-s − 23.4·16-s + 745.·17-s − 286.·18-s + 1.29e3·19-s + 1.79e3·20-s − 80.9·21-s − 2.17e3·22-s − 529·23-s + 1.64e3·24-s + 5.36e3·25-s + 1.75e3·26-s + 729·27-s + 174.·28-s + ⋯
 L(s)  = 1 − 0.626·2-s + 0.577·3-s − 0.607·4-s − 1.64·5-s − 0.361·6-s − 0.0693·7-s + 1.00·8-s + 0.333·9-s + 1.03·10-s + 1.52·11-s − 0.350·12-s − 0.814·13-s + 0.0434·14-s − 0.951·15-s − 0.0228·16-s + 0.625·17-s − 0.208·18-s + 0.825·19-s + 1.00·20-s − 0.0400·21-s − 0.956·22-s − 0.208·23-s + 0.581·24-s + 1.71·25-s + 0.510·26-s + 0.192·27-s + 0.0421·28-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$69$$    =    $$3 \cdot 23$$ Sign: $1$ Analytic conductor: $$11.0664$$ Root analytic conductor: $$3.32663$$ Motivic weight: $$5$$ Rational: no Arithmetic: yes Character: $\chi_{69} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 69,\ (\ :5/2),\ 1)$$

## Particular Values

 $$L(3)$$ $$\approx$$ $$0.9664536085$$ $$L(\frac12)$$ $$\approx$$ $$0.9664536085$$ $$L(\frac{7}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 9T$$
23 $$1 + 529T$$
good2 $$1 + 3.54T + 32T^{2}$$
5 $$1 + 92.1T + 3.12e3T^{2}$$
7 $$1 + 8.98T + 1.68e4T^{2}$$
11 $$1 - 612.T + 1.61e5T^{2}$$
13 $$1 + 496.T + 3.71e5T^{2}$$
17 $$1 - 745.T + 1.41e6T^{2}$$
19 $$1 - 1.29e3T + 2.47e6T^{2}$$
29 $$1 - 7.75e3T + 2.05e7T^{2}$$
31 $$1 + 1.06e3T + 2.86e7T^{2}$$
37 $$1 + 840.T + 6.93e7T^{2}$$
41 $$1 - 9.82e3T + 1.15e8T^{2}$$
43 $$1 + 1.35e4T + 1.47e8T^{2}$$
47 $$1 - 2.04e4T + 2.29e8T^{2}$$
53 $$1 - 4.81e3T + 4.18e8T^{2}$$
59 $$1 - 3.05e4T + 7.14e8T^{2}$$
61 $$1 - 9.53T + 8.44e8T^{2}$$
67 $$1 - 9.99e3T + 1.35e9T^{2}$$
71 $$1 + 7.37e4T + 1.80e9T^{2}$$
73 $$1 - 5.51e4T + 2.07e9T^{2}$$
79 $$1 - 6.23e4T + 3.07e9T^{2}$$
83 $$1 + 8.24e4T + 3.93e9T^{2}$$
89 $$1 - 9.51e4T + 5.58e9T^{2}$$
97 $$1 - 4.71e4T + 8.58e9T^{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

−14.02444355149951149352689985262, −12.42782918073078516525213616656, −11.63122173781117147315355718384, −10.05818148688829285093997454709, −8.985329741924781868167586931587, −8.037280753225211284468601151860, −7.12167511553267171037756707280, −4.57896778918635400770656627339, −3.53511996991014255088625359725, −0.860939740144385835086401300255, 0.860939740144385835086401300255, 3.53511996991014255088625359725, 4.57896778918635400770656627339, 7.12167511553267171037756707280, 8.037280753225211284468601151860, 8.985329741924781868167586931587, 10.05818148688829285093997454709, 11.63122173781117147315355718384, 12.42782918073078516525213616656, 14.02444355149951149352689985262