# Properties

 Label 2-1216-152.21-c0-0-0 Degree $2$ Conductor $1216$ Sign $0.959 + 0.281i$ Analytic cond. $0.606863$ Root an. cond. $0.779014$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (−1.50 − 1.26i)3-s + (0.500 + 2.83i)9-s + (1.32 + 0.766i)11-s + (−0.173 + 0.984i)17-s + (−0.342 + 0.939i)19-s + (0.766 − 0.642i)25-s + (1.85 − 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (0.342 + 0.939i)43-s + (0.5 − 0.866i)49-s + (1.50 − 1.26i)51-s + (1.70 − 0.984i)57-s + (−0.223 + 1.26i)59-s + (−0.118 − 0.673i)67-s + ⋯
 L(s)  = 1 + (−1.50 − 1.26i)3-s + (0.500 + 2.83i)9-s + (1.32 + 0.766i)11-s + (−0.173 + 0.984i)17-s + (−0.342 + 0.939i)19-s + (0.766 − 0.642i)25-s + (1.85 − 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (0.342 + 0.939i)43-s + (0.5 − 0.866i)49-s + (1.50 − 1.26i)51-s + (1.70 − 0.984i)57-s + (−0.223 + 1.26i)59-s + (−0.118 − 0.673i)67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1216$$    =    $$2^{6} \cdot 19$$ Sign: $0.959 + 0.281i$ Analytic conductor: $$0.606863$$ Root analytic conductor: $$0.779014$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1216} (97, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1216,\ (\ :0),\ 0.959 + 0.281i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6694147326$$ $$L(\frac12)$$ $$\approx$$ $$0.6694147326$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
19 $$1 + (0.342 - 0.939i)T$$
good3 $$1 + (1.50 + 1.26i)T + (0.173 + 0.984i)T^{2}$$
5 $$1 + (-0.766 + 0.642i)T^{2}$$
7 $$1 + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-1.32 - 0.766i)T + (0.5 + 0.866i)T^{2}$$
13 $$1 + (0.173 - 0.984i)T^{2}$$
17 $$1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}$$
23 $$1 + (0.766 + 0.642i)T^{2}$$
29 $$1 + (-0.939 + 0.342i)T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 + T^{2}$$
41 $$1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2}$$
43 $$1 + (-0.342 - 0.939i)T + (-0.766 + 0.642i)T^{2}$$
47 $$1 + (-0.939 + 0.342i)T^{2}$$
53 $$1 + (0.766 + 0.642i)T^{2}$$
59 $$1 + (0.223 - 1.26i)T + (-0.939 - 0.342i)T^{2}$$
61 $$1 + (-0.766 - 0.642i)T^{2}$$
67 $$1 + (0.118 + 0.673i)T + (-0.939 + 0.342i)T^{2}$$
71 $$1 + (-0.766 + 0.642i)T^{2}$$
73 $$1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2}$$
79 $$1 + (-0.173 - 0.984i)T^{2}$$
83 $$1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2}$$
89 $$1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2}$$
97 $$1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2}$$
show more
show less
$$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

−10.27641339167269573519435032129, −8.998993558701019896141797040433, −7.979884364153337119762600971611, −7.24139452214718781095818405374, −6.38263135896529070721597018102, −6.08400663843026785017736672166, −4.92938352888714180568780801509, −4.05527060704066262215472225775, −2.12104111770290365155101052704, −1.23910926268359474326035321563, 0.874569412338624668585024776104, 3.16764822558467111678555864723, 4.12457433277935380765076252523, 4.84529431911140568271770440989, 5.68425049102192360720198570067, 6.47199588614834529112290557857, 7.10953149754410534283276758894, 8.857736404653668686460676085550, 9.215085877188752380496596817711, 10.02036443937041995306462382893