↑

Properties

Label 2-224-56.53-c1-0-0
Degree $2$
Conductor $224$
Sign $0.280 - 0.959i$
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  + (−2.13 − 1.22i)3-s + (−1.28 + 0.742i)5-s + (−0.129 + 2.64i)7-s + (1.52 + 2.64i)9-s + (4.37 + 2.52i)11-s + 2.58i·13-s + 3.65·15-s + (−0.629 + 1.09i)17-s + (−2.68 + 1.54i)19-s + (3.52 − 5.47i)21-s + (−0.697 − 1.20i)23-s + (−1.39 + 2.41i)25-s − 0.126i·27-s − 0.638i·29-s + (−1.82 + 3.16i)31-s + ⋯
L(s)  = 1  + (−1.22 − 0.710i)3-s + (−0.575 + 0.332i)5-s + (−0.0490 + 0.998i)7-s + (0.508 + 0.880i)9-s + (1.31 + 0.760i)11-s + 0.717i·13-s + 0.943·15-s + (−0.152 + 0.264i)17-s + (−0.615 + 0.355i)19-s + (0.769 − 1.19i)21-s + (−0.145 − 0.252i)23-s + (−0.279 + 0.483i)25-s − 0.0243i·27-s − 0.118i·29-s + (−0.328 + 0.568i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.280 - 0.959i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.280 - 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.486781 + 0.364822i\)
\(L(\frac12)\) \(\approx\) \(0.486781 + 0.364822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.129 - 2.64i)T \)
good3 \( 1 + (2.13 + 1.22i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.28 - 0.742i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.37 - 2.52i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.58iT - 13T^{2} \)
17 \( 1 + (0.629 - 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.697 + 1.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.638iT - 29T^{2} \)
31 \( 1 + (1.82 - 3.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.21 - 3.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.36T + 41T^{2} \)
43 \( 1 + 1.02iT - 43T^{2} \)
47 \( 1 + (5.48 + 9.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.99 - 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.01 - 1.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.1 + 6.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.443 + 0.256i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.41T + 71T^{2} \)
73 \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.35 - 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.97iT - 83T^{2} \)
89 \( 1 + (-1.29 - 2.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.57T + 97T^{2} \)
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

−12.07296322074608178403398928780, −11.81090787632649219401668667663, −10.90292037646279887918416959655, −9.550463300984065431362168004137, −8.462350948291587324703643849915, −7.00234316929147723983249008319, −6.50390441287699946004958278039, −5.36402017015951605085208621590, −3.95939416872215830529147323441, −1.82471537950692023762108939149, 0.60843294209001267401494823743, 3.74847564408547277642612560083, 4.51561503782930988384248410072, 5.78404146536148031188477763220, 6.77529864523697320885807796833, 8.068691270275096510294267537417, 9.313644209733722783598992069621, 10.38562068481014179250448950452, 11.14698719339297917894662945584, 11.74707624304789255728967070121

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