# Properties

 Label 2-12-1.1-c21-0-2 Degree $2$ Conductor $12$ Sign $-1$ Analytic cond. $33.5372$ Root an. cond. $5.79113$ Motivic weight $21$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

# Origins

## Dirichlet series

 L(s)  = 1 + 5.90e4·3-s − 1.12e7·5-s + 2.81e8·7-s + 3.48e9·9-s − 3.61e10·11-s − 4.49e11·13-s − 6.65e11·15-s + 2.12e12·17-s − 4.60e12·19-s + 1.66e13·21-s + 9.50e13·23-s − 3.49e14·25-s + 2.05e14·27-s − 2.24e15·29-s − 3.15e15·31-s − 2.13e15·33-s − 3.17e15·35-s − 1.81e16·37-s − 2.65e16·39-s − 1.69e17·41-s − 1.58e17·43-s − 3.92e16·45-s − 1.34e17·47-s − 4.79e17·49-s + 1.25e17·51-s − 1.56e16·53-s + 4.07e17·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.516·5-s + 0.377·7-s + 1/3·9-s − 0.420·11-s − 0.903·13-s − 0.297·15-s + 0.255·17-s − 0.172·19-s + 0.217·21-s + 0.478·23-s − 0.733·25-s + 0.192·27-s − 0.991·29-s − 0.691·31-s − 0.242·33-s − 0.194·35-s − 0.621·37-s − 0.521·39-s − 1.97·41-s − 1.12·43-s − 0.172·45-s − 0.373·47-s − 0.857·49-s + 0.147·51-s − 0.0122·53-s + 0.216·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$12$$    =    $$2^{2} \cdot 3$$ Sign: $-1$ Analytic conductor: $$33.5372$$ Root analytic conductor: $$5.79113$$ Motivic weight: $$21$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 12,\ (\ :21/2),\ -1)$$

## Particular Values

 $$L(11)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{23}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - p^{10} T$$
good5 $$1 + 2253618 p T + p^{21} T^{2}$$
7 $$1 - 40273448 p T + p^{21} T^{2}$$
11 $$1 + 36172082484 T + p^{21} T^{2}$$
13 $$1 + 34546044490 p T + p^{21} T^{2}$$
17 $$1 - 124815222738 p T + p^{21} T^{2}$$
19 $$1 + 242600328100 p T + p^{21} T^{2}$$
23 $$1 - 95095276921656 T + p^{21} T^{2}$$
29 $$1 + 77439392529354 p T + p^{21} T^{2}$$
31 $$1 + 3155693201792656 T + p^{21} T^{2}$$
37 $$1 + 18178503074861482 T + p^{21} T^{2}$$
41 $$1 + 169649739387485910 T + p^{21} T^{2}$$
43 $$1 + 158968551608988244 T + p^{21} T^{2}$$
47 $$1 + 134697468442682736 T + p^{21} T^{2}$$
53 $$1 + 15637375269722538 T + p^{21} T^{2}$$
59 $$1 - 2977241337691499484 T + p^{21} T^{2}$$
61 $$1 - 3603855625679330702 T + p^{21} T^{2}$$
67 $$1 - 21066199531967164004 T + p^{21} T^{2}$$
71 $$1 - 21980089544074358760 T + p^{21} T^{2}$$
73 $$1 + 17054415965500339222 T + p^{21} T^{2}$$
79 $$1 +$$$$11\!\cdots\!52$$$$T + p^{21} T^{2}$$
83 $$1 + 96628520442403345644 T + p^{21} T^{2}$$
89 $$1 - 60427571095732966650 T + p^{21} T^{2}$$
97 $$1 +$$$$40\!\cdots\!98$$$$T + p^{21} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$