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Properties

Label 2-224-224.13-c2-0-21
Degree $2$
Conductor $224$
Sign $0.682 - 0.730i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.62 − 1.16i)2-s + (3.33 + 1.37i)3-s + (1.28 + 3.78i)4-s + (3.52 + 8.49i)5-s + (−3.80 − 6.12i)6-s + (6.86 − 1.37i)7-s + (2.32 − 7.65i)8-s + (2.82 + 2.82i)9-s + (4.17 − 17.9i)10-s + (−3.28 − 7.94i)11-s + (−0.941 + 14.3i)12-s + (−7.03 + 16.9i)13-s + (−12.7 − 5.76i)14-s + 33.1i·15-s + (−12.6 + 9.74i)16-s − 0.600·17-s + ⋯
L(s)  = 1  + (−0.812 − 0.582i)2-s + (1.11 + 0.459i)3-s + (0.321 + 0.946i)4-s + (0.704 + 1.69i)5-s + (−0.634 − 1.02i)6-s + (0.980 − 0.196i)7-s + (0.290 − 0.956i)8-s + (0.313 + 0.313i)9-s + (0.417 − 1.79i)10-s + (−0.299 − 0.722i)11-s + (−0.0784 + 1.19i)12-s + (−0.541 + 1.30i)13-s + (−0.911 − 0.411i)14-s + 2.21i·15-s + (−0.793 + 0.608i)16-s − 0.0353·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.682 - 0.730i$
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: \((2,\ 224,\ (\ :1),\ 0.682 - 0.730i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60660 + 0.697730i\)
\(L(\frac12)\) \(\approx\) \(1.60660 + 0.697730i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.62 + 1.16i)T \)
7 \( 1 + (-6.86 + 1.37i)T \)
good3 \( 1 + (-3.33 - 1.37i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-3.52 - 8.49i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (3.28 + 7.94i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (7.03 - 16.9i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 0.600T + 289T^{2} \)
19 \( 1 + (-3.14 + 7.59i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (0.165 + 0.165i)T + 529iT^{2} \)
29 \( 1 + (-14.5 + 35.0i)T + (-594. - 594. i)T^{2} \)
31 \( 1 + 21.7iT - 961T^{2} \)
37 \( 1 + (-30.4 + 12.5i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (15.4 - 15.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (-28.0 - 67.6i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 23.9T + 2.20e3T^{2} \)
53 \( 1 + (19.4 + 47.0i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-13.0 - 31.5i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-51.4 - 21.3i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-38.4 + 92.9i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-56.8 + 56.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (86.4 - 86.4i)T - 5.32e3iT^{2} \)
79 \( 1 - 134. iT - 6.24e3T^{2} \)
83 \( 1 + (-23.6 + 56.9i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (87.5 + 87.5i)T + 7.92e3iT^{2} \)
97 \( 1 + 49.2iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47777841657036630461945789641, −11.19161080389979509359580726923, −9.986745431931062580489579803531, −9.491495533549610228186115417320, −8.316283728151826347177288230968, −7.46632529575799844615841967395, −6.35592563640999373930750471090, −4.15736900550492134611982690300, −2.88929004421373763654470258693, −2.15040737274385191701836870724, 1.23560365591636160902538002003, 2.28332300448119668641696147070, 4.92026203309543495334628548988, 5.51606517809635470449418674064, 7.35735054387089916622178250641, 8.204942664056029612133064997744, 8.648346850849394729925092555375, 9.560550464370772947146800231328, 10.51417862526019159710155451637, 12.15888111086496437541207458055

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