↑

Properties

Label 2-224-56.37-c1-0-2
Degree $2$
Conductor $224$
Sign $0.906 - 0.422i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.591 − 0.341i)3-s + (2.80 + 1.61i)5-s + (−1.47 + 2.19i)7-s + (−1.26 + 2.19i)9-s + (2.08 − 1.20i)11-s − 3.09i·13-s + 2.21·15-s + (−1.97 − 3.42i)17-s + (2.33 + 1.35i)19-s + (−0.124 + 1.80i)21-s + (1.37 − 2.37i)23-s + (2.74 + 4.75i)25-s + 3.77i·27-s − 2.01i·29-s + (−1.10 − 1.91i)31-s + ⋯
L(s)  = 1  + (0.341 − 0.197i)3-s + (1.25 + 0.724i)5-s + (−0.558 + 0.829i)7-s + (−0.422 + 0.731i)9-s + (0.629 − 0.363i)11-s − 0.858i·13-s + 0.570·15-s + (−0.479 − 0.830i)17-s + (0.536 + 0.309i)19-s + (−0.0271 + 0.393i)21-s + (0.286 − 0.495i)23-s + (0.548 + 0.950i)25-s + 0.726i·27-s − 0.374i·29-s + (−0.198 − 0.343i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.906 - 0.422i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47847 + 0.328045i\)
\(L(\frac12)\) \(\approx\) \(1.47847 + 0.328045i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.47 - 2.19i)T \)
good3 \( 1 + (-0.591 + 0.341i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + (1.97 + 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.33 - 1.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.37 + 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (1.10 + 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.30 + 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 + 11.5iT - 43T^{2} \)
47 \( 1 + (3.31 - 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.23 - 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.6 - 6.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.54 - 4.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.01 + 2.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.64T + 71T^{2} \)
73 \( 1 + (-4.77 - 8.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.838 + 1.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.47iT - 83T^{2} \)
89 \( 1 + (6.98 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.43940453837119180978950506502, −11.26738092907470183567206127702, −10.30530695367739745914968740972, −9.395794381005226628708673405939, −8.550477616993866786857715052659, −7.17727584172984609359909634302, −6.08767237478240573191770898355, −5.32784867779354748793925265292, −3.12439404643894399118371349480, −2.23148025960016246201873238520, 1.57279958854662576055538365366, 3.44757919603687801137550330122, 4.72001591879464146835558390140, 6.13467110964246293747059583558, 6.89779773275639917925910169520, 8.570700973834039598931046387850, 9.433777273298068725922302700610, 9.840601111276603547713022165240, 11.18589077416528941293695852312, 12.36996969760791784822137924074

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