Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7 $
Sign $0.777 - 0.628i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 − 0.601i)3-s + (0.0726 − 0.125i)5-s + (−1.05 + 2.42i)7-s + (2.27 + 1.95i)9-s + (2.13 − 1.23i)11-s + 2.04i·13-s + (−0.193 + 0.160i)15-s + (0.878 + 1.52i)17-s + (3.68 + 2.12i)19-s + (3.17 − 3.30i)21-s + (7.46 + 4.30i)23-s + (2.48 + 4.31i)25-s + (−2.52 − 4.54i)27-s − 7.08i·29-s + (−3.11 + 1.80i)31-s + ⋯
L(s)  = 1  + (−0.937 − 0.347i)3-s + (0.0324 − 0.0562i)5-s + (−0.398 + 0.917i)7-s + (0.758 + 0.651i)9-s + (0.644 − 0.372i)11-s + 0.566i·13-s + (−0.0500 + 0.0414i)15-s + (0.213 + 0.369i)17-s + (0.844 + 0.487i)19-s + (0.692 − 0.721i)21-s + (1.55 + 0.898i)23-s + (0.497 + 0.862i)25-s + (−0.485 − 0.874i)27-s − 1.31i·29-s + (−0.560 + 0.323i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
\( \varepsilon \)  =  $0.777 - 0.628i$
motivic weight  =  \(1\)
character  :  $\chi_{336} (257, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 336,\ (\ :1/2),\ 0.777 - 0.628i)$
$L(1)$  $\approx$  $0.905632 + 0.320019i$
$L(\frac12)$  $\approx$  $0.905632 + 0.320019i$
$L(\frac{3}{2})$   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 \( 1 + (1.62 + 0.601i)T \)
7 \( 1 + (1.05 - 2.42i)T \)
good5 \( 1 + (-0.0726 + 0.125i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.13 + 1.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.04iT - 13T^{2} \)
17 \( 1 + (-0.878 - 1.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.68 - 2.12i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.46 - 4.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.08iT - 29T^{2} \)
31 \( 1 + (3.11 - 1.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.93 - 5.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.33T + 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 + (4.65 - 8.05i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.49 - 2.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.60 + 9.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.66 - 2.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.57 + 4.45i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.79iT - 71T^{2} \)
73 \( 1 + (-11.3 + 6.52i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.86 - 4.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + (-4.34 + 7.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.65iT - 97T^{2} \)
show more
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

−11.63035121668110752710272205683, −11.04692715637189193696923449177, −9.724695741560368334091801764939, −9.033569511648807992475374782158, −7.71095664105772249725588638682, −6.65532308426650568938715832112, −5.83594444195217208183824095008, −4.92444211045043322259119631627, −3.33086233561778549727278599616, −1.49654499229327281141850042649, 0.873019747436098907610337148180, 3.27741344146979881104826577987, 4.48946921205475530609961985357, 5.43752259464543677414530703938, 6.79718401553424919272954664801, 7.19475106883633411383256895356, 8.872966457190846339568835725032, 9.809732690775915167685264052036, 10.60825548535164878521677387060, 11.26828170973407123366093983687

Graph of the $Z$-function along the critical line