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Properties

Label 2-224-56.5-c2-0-11
Degree $2$
Conductor $224$
Sign $-0.624 + 0.780i$
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  + (−1.94 − 3.37i)3-s + (4.42 − 7.67i)5-s + (6.92 − 1.03i)7-s + (−3.09 + 5.35i)9-s + (−3.15 + 1.82i)11-s + 7.79·13-s − 34.5·15-s + (−9.07 + 5.23i)17-s + (5.39 − 9.34i)19-s + (−16.9 − 21.3i)21-s + (−6.45 + 11.1i)23-s + (−26.7 − 46.3i)25-s − 10.9·27-s + 17.2i·29-s + (26.1 − 15.1i)31-s + ⋯
L(s)  = 1  + (−0.649 − 1.12i)3-s + (0.885 − 1.53i)5-s + (0.989 − 0.147i)7-s + (−0.343 + 0.594i)9-s + (−0.287 + 0.165i)11-s + 0.599·13-s − 2.30·15-s + (−0.533 + 0.308i)17-s + (0.283 − 0.491i)19-s + (−0.808 − 1.01i)21-s + (−0.280 + 0.486i)23-s + (−1.06 − 1.85i)25-s − 0.406·27-s + 0.594i·29-s + (0.844 − 0.487i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.624 + 0.780i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.661117 - 1.37551i\)
\(L(\frac12)\) \(\approx\) \(0.661117 - 1.37551i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-6.92 + 1.03i)T \)
good3 \( 1 + (1.94 + 3.37i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-4.42 + 7.67i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (3.15 - 1.82i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 - 7.79T + 169T^{2} \)
17 \( 1 + (9.07 - 5.23i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-5.39 + 9.34i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (6.45 - 11.1i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 17.2iT - 841T^{2} \)
31 \( 1 + (-26.1 + 15.1i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (34.2 + 19.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 73.6iT - 1.68e3T^{2} \)
43 \( 1 - 40.8iT - 1.84e3T^{2} \)
47 \( 1 + (-36.2 - 20.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-5.55 + 3.20i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (7.95 + 13.7i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.07 + 10.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.75 + 3.89i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + (-77.6 + 44.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-35.3 + 61.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 60.8T + 6.88e3T^{2} \)
89 \( 1 + (23.4 + 13.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 3.26iT - 9.40e3T^{2} \)
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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

−11.86707845884149753105343448118, −10.98668745318946965677085103991, −9.609652930638876376723764908214, −8.577349323784700161664620495260, −7.73864893410059294085698452331, −6.39664767091496698404825910258, −5.47100669579471356729207350917, −4.56322378874166447783519202268, −1.87214536882345445114836127537, −0.983074735940171440461394743583, 2.26656177277850376116851572860, 3.80009989188494090503579958858, 5.17626284095366537817463498566, 6.00281470913398900527391530365, 7.15472423183315861778585005969, 8.586579894547197736682404190938, 9.878495882629170372643050906789, 10.56253042167163769563286756225, 11.02359923871252581805124108751, 11.95954374500140801092200697574

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