Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.487 + 0.872i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.125 + 1.99i)2-s + (−3.96 + 0.5i)4-s + (−4.94 − 4.94i)7-s + (−1.49 − 7.85i)8-s + (6.36 − 6.36i)9-s + (−15.9 − 6.60i)11-s + (9.26 − 10.5i)14-s + (15.5 − 3.96i)16-s + (13.5 + 11.9i)18-s + (11.1 − 32.6i)22-s + (12.1 − 12.1i)23-s + (−17.6 − 17.6i)25-s + (22.1 + 17.1i)28-s + (−1.02 + 0.424i)29-s + (9.86 + 30.4i)32-s + ⋯
L(s)  = 1  + (0.0626 + 0.998i)2-s + (−0.992 + 0.125i)4-s + (−0.707 − 0.707i)7-s + (−0.186 − 0.982i)8-s + (0.707 − 0.707i)9-s + (−1.44 − 0.600i)11-s + (0.661 − 0.750i)14-s + (0.968 − 0.248i)16-s + (0.750 + 0.661i)18-s + (0.508 − 1.48i)22-s + (0.528 − 0.528i)23-s + (−0.707 − 0.707i)25-s + (0.789 + 0.613i)28-s + (−0.0353 + 0.0146i)29-s + (0.308 + 0.951i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.487 + 0.872i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (181, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.487 + 0.872i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.648894 - 0.380689i\)
\(L(\frac12)\)  \(\approx\)  \(0.648894 - 0.380689i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.125 - 1.99i)T \)
7 \( 1 + (4.94 + 4.94i)T \)
good3 \( 1 + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (17.6 + 17.6i)T^{2} \)
11 \( 1 + (15.9 + 6.60i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (119. - 119. i)T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + (255. - 255. i)T^{2} \)
23 \( 1 + (-12.1 + 12.1i)T - 529iT^{2} \)
29 \( 1 + (1.02 - 0.424i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + (-4.13 + 9.98i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + 1.68e3iT^{2} \)
43 \( 1 + (71.9 + 29.8i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + (-42.5 - 17.6i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (78.2 - 32.4i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (59.8 + 59.8i)T + 5.04e3iT^{2} \)
73 \( 1 + 5.32e3iT^{2} \)
79 \( 1 + 23.3iT - 6.24e3T^{2} \)
83 \( 1 + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 - 7.92e3iT^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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

−12.19553317253624494142732979385, −10.52394983674406398763184555506, −9.897987526494636909810255760086, −8.745618039077534390583994723315, −7.67439292378974322935058394773, −6.81205442811634421860088437260, −5.81870930452265543576691763452, −4.50614807743334454131192218755, −3.30547028396814748049010646527, −0.39628387631228295883303581626, 1.97881038624140451074088826742, 3.16097539221147259327103074883, 4.71807783612912724964773627436, 5.61539814466611701251673560671, 7.34192557039458790907914437520, 8.415151474246194817512860635872, 9.660027849655339342706564361692, 10.17943306257337627746410008208, 11.21670413430389434035646709861, 12.24887919947692616509894018939

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