# Properties

 Degree 2 Conductor $2^{2} \cdot 3$ Sign $-0.0359 - 0.999i$ Motivic weight 19 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (723. + 17.4i)2-s + (−2.86e3 − 3.39e4i)3-s + (5.23e5 + 2.52e4i)4-s + 5.49e6i·5-s + (−1.48e6 − 2.46e7i)6-s + 1.05e8i·7-s + (3.78e8 + 2.74e7i)8-s + (−1.14e9 + 1.94e8i)9-s + (−9.57e7 + 3.97e9i)10-s − 7.09e9·11-s + (−6.43e8 − 1.78e10i)12-s − 6.63e10·13-s + (−1.83e9 + 7.61e10i)14-s + (1.86e11 − 1.57e10i)15-s + (2.73e11 + 2.64e10i)16-s + 6.50e11i·17-s + ⋯
 L(s)  = 1 + (0.999 + 0.0240i)2-s + (−0.0840 − 0.996i)3-s + (0.998 + 0.0481i)4-s + 1.25i·5-s + (−0.0600 − 0.998i)6-s + 0.985i·7-s + (0.997 + 0.0721i)8-s + (−0.985 + 0.167i)9-s + (−0.0302 + 1.25i)10-s − 0.907·11-s + (−0.0359 − 0.999i)12-s − 1.73·13-s + (−0.0237 + 0.985i)14-s + (1.25 − 0.105i)15-s + (0.995 + 0.0961i)16-s + 1.33i·17-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(20-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.0359 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$12$$    =    $$2^{2} \cdot 3$$ $$\varepsilon$$ = $-0.0359 - 0.999i$ motivic weight = $$19$$ character : $\chi_{12} (11, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 12,\ (\ :19/2),\ -0.0359 - 0.999i)$ $L(10)$ $\approx$ $2.511793317$ $L(\frac12)$ $\approx$ $2.511793317$ $L(\frac{21}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1 + (-723. - 17.4i)T$$
3 $$1 + (2.86e3 + 3.39e4i)T$$
good5 $$1 - 5.49e6iT - 1.90e13T^{2}$$
7 $$1 - 1.05e8iT - 1.13e16T^{2}$$
11 $$1 + 7.09e9T + 6.11e19T^{2}$$
13 $$1 + 6.63e10T + 1.46e21T^{2}$$
17 $$1 - 6.50e11iT - 2.39e23T^{2}$$
19 $$1 - 8.23e11iT - 1.97e24T^{2}$$
23 $$1 - 1.08e13T + 7.46e25T^{2}$$
29 $$1 + 9.35e13iT - 6.10e27T^{2}$$
31 $$1 - 1.40e14iT - 2.16e28T^{2}$$
37 $$1 - 2.24e14T + 6.24e29T^{2}$$
41 $$1 - 3.53e14iT - 4.39e30T^{2}$$
43 $$1 + 1.91e15iT - 1.08e31T^{2}$$
47 $$1 + 1.54e15T + 5.88e31T^{2}$$
53 $$1 + 2.30e16iT - 5.77e32T^{2}$$
59 $$1 + 7.28e16T + 4.42e33T^{2}$$
61 $$1 + 3.87e16T + 8.34e33T^{2}$$
67 $$1 - 3.26e17iT - 4.95e34T^{2}$$
71 $$1 - 1.37e17T + 1.49e35T^{2}$$
73 $$1 - 2.84e17T + 2.53e35T^{2}$$
79 $$1 - 3.04e17iT - 1.13e36T^{2}$$
83 $$1 - 1.54e18T + 2.90e36T^{2}$$
89 $$1 + 1.94e18iT - 1.09e37T^{2}$$
97 $$1 - 3.32e17T + 5.60e37T^{2}$$
show less
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−15.18127939745174999591918879147, −14.49069574167572928254882692523, −12.93013170541104013177315196612, −11.94757367999841011542038499797, −10.55961924000322895225127515283, −7.80144387217403824138449101198, −6.62940910757103460984632056394, −5.38308800567532671085685450568, −2.91876457177305249631246779087, −2.14629629828158637412255930112, 0.53829853040758446945923904512, 2.84640465004266078888375689197, 4.67369516887414929662770744336, 5.05923229654120571178921634779, 7.42666816342303112925832226979, 9.502454265134908711528094766240, 10.92580688451190492750771804594, 12.42732532320741410637931783499, 13.70347583491791151750332740495, 15.11509165903866829357980687873