↑

Properties

Label 2-224-224.13-c2-0-3
Degree $2$
Conductor $224$
Sign $0.120 - 0.992i$
Analytic cond. $6.10355$
Root an. cond. $2.47053$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.28 − 1.53i)2-s + (−0.416 − 0.172i)3-s + (−0.709 + 3.93i)4-s + (1.21 + 2.94i)5-s + (0.269 + 0.861i)6-s + (−3.48 − 6.07i)7-s + (6.95 − 3.96i)8-s + (−6.22 − 6.22i)9-s + (2.95 − 5.64i)10-s + (6.49 + 15.6i)11-s + (0.975 − 1.51i)12-s + (−3.18 + 7.67i)13-s + (−4.84 + 13.1i)14-s − 1.43i·15-s + (−14.9 − 5.58i)16-s − 19.7·17-s + ⋯
L(s)  = 1  + (−0.641 − 0.767i)2-s + (−0.138 − 0.0575i)3-s + (−0.177 + 0.984i)4-s + (0.243 + 0.588i)5-s + (0.0449 + 0.143i)6-s + (−0.497 − 0.867i)7-s + (0.868 − 0.495i)8-s + (−0.691 − 0.691i)9-s + (0.295 − 0.564i)10-s + (0.590 + 1.42i)11-s + (0.0812 − 0.126i)12-s + (−0.244 + 0.590i)13-s + (−0.346 + 0.938i)14-s − 0.0957i·15-s + (−0.937 − 0.349i)16-s − 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.120 - 0.992i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.120 - 0.992i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.387118 + 0.342814i\)
\(L(\frac12)\) \(\approx\) \(0.387118 + 0.342814i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 1.53i)T \)
7 \( 1 + (3.48 + 6.07i)T \)
good3 \( 1 + (0.416 + 0.172i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-1.21 - 2.94i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-6.49 - 15.6i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (3.18 - 7.67i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 19.7T + 289T^{2} \)
19 \( 1 + (12.0 - 29.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (-7.09 - 7.09i)T + 529iT^{2} \)
29 \( 1 + (6.56 - 15.8i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 57.4iT - 961T^{2} \)
37 \( 1 + (-1.99 + 0.828i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-10.4 + 10.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (6.95 + 16.7i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 58.0T + 2.20e3T^{2} \)
53 \( 1 + (8.74 + 21.1i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (28.6 + 69.0i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-24.1 - 10.0i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (42.9 - 103. i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-44.2 + 44.2i)T - 5.04e3iT^{2} \)
73 \( 1 + (-7.91 + 7.91i)T - 5.32e3iT^{2} \)
79 \( 1 - 67.3iT - 6.24e3T^{2} \)
83 \( 1 + (-20.0 + 48.4i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (88.5 + 88.5i)T + 7.92e3iT^{2} \)
97 \( 1 + 172. iT - 9.40e3T^{2} \)
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

−12.20247258700392913800849387622, −11.13985660972329960794548126089, −10.28489556728832756652129165858, −9.554104678117699880430931140902, −8.574165286878668877204349305513, −7.05126786749347571980139790221, −6.61726039815846437939708402896, −4.43971070102024320104388899498, −3.33752898047550085418070702415, −1.76927148162630543750060235536, 0.34310038924711438813793516767, 2.53511235906497310808523874181, 4.78211417019180939391650532764, 5.78364976957521513224051495316, 6.51044438728853510298043137241, 8.091827007182422278547218252627, 8.860500776786815393165433145150, 9.392375288928622672027461123779, 10.91033143864250999606327984085, 11.42727362302634508583221951765

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