# Properties

 Degree 2 Conductor $2^{4} \cdot 3 \cdot 7$ Sign $-0.304 - 0.952i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + (1.45 + 0.934i)3-s + (−1.90 + 3.29i)5-s + (−2.23 − 1.41i)7-s + (1.25 + 2.72i)9-s + (0.309 − 0.178i)11-s + 4.04i·13-s + (−5.84 + 3.02i)15-s + (−0.0519 − 0.0900i)17-s + (2.12 + 1.22i)19-s + (−1.93 − 4.15i)21-s + (−1.15 − 0.665i)23-s + (−4.72 − 8.17i)25-s + (−0.723 + 5.14i)27-s + 4.97i·29-s + (6.83 − 3.94i)31-s + ⋯
 L(s)  = 1 + (0.841 + 0.539i)3-s + (−0.849 + 1.47i)5-s + (−0.844 − 0.535i)7-s + (0.417 + 0.908i)9-s + (0.0933 − 0.0538i)11-s + 1.12i·13-s + (−1.50 + 0.780i)15-s + (−0.0126 − 0.0218i)17-s + (0.487 + 0.281i)19-s + (−0.422 − 0.906i)21-s + (−0.240 − 0.138i)23-s + (−0.944 − 1.63i)25-s + (−0.139 + 0.990i)27-s + 0.923i·29-s + (1.22 − 0.708i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$336$$    =    $$2^{4} \cdot 3 \cdot 7$$ $$\varepsilon$$ = $-0.304 - 0.952i$ motivic weight = $$1$$ character : $\chi_{336} (257, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 336,\ (\ :1/2),\ -0.304 - 0.952i)$ $L(1)$ $\approx$ $0.769182 + 1.05316i$ $L(\frac12)$ $\approx$ $0.769182 + 1.05316i$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-1.45 - 0.934i)T$$
7 $$1 + (2.23 + 1.41i)T$$
good5 $$1 + (1.90 - 3.29i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (-0.309 + 0.178i)T + (5.5 - 9.52i)T^{2}$$
13 $$1 - 4.04iT - 13T^{2}$$
17 $$1 + (0.0519 + 0.0900i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (-2.12 - 1.22i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (1.15 + 0.665i)T + (11.5 + 19.9i)T^{2}$$
29 $$1 - 4.97iT - 29T^{2}$$
31 $$1 + (-6.83 + 3.94i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 + (-5.45 + 9.45i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 - 6.15T + 41T^{2}$$
43 $$1 + 0.502T + 43T^{2}$$
47 $$1 + (5.72 - 9.91i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-5.08 + 2.93i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + (3.77 + 6.53i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-8.20 - 4.73i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-1.34 - 2.32i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + 5.78iT - 71T^{2}$$
73 $$1 + (0.203 - 0.117i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (-1.61 + 2.79i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 - 9.07T + 83T^{2}$$
89 $$1 + (-3.41 + 5.90i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + 5.14iT - 97T^{2}$$
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\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

−11.55279525522866921362255347803, −10.82972185319706724503133838808, −9.982538972809937611340927580644, −9.214167886608814393697660751836, −7.88700365179089065986067197980, −7.19508377831891081668701020108, −6.30327243512666475269033143958, −4.31090868008685798545777892672, −3.57692607484081581047623259910, −2.59526906863854710148238366343, 0.871030348033655795552330098128, 2.81393952623480763804756503467, 3.96995128990737407099586955054, 5.25127195187350083973448930200, 6.52808451544070357470227451900, 7.85477903362729193107898077611, 8.336009866331646022460798534154, 9.232027856707322671390029697869, 10.00833320195499414705542666606, 11.79645280889782524004423416912