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Properties

Label 2-224-28.23-c2-0-13
Degree $2$
Conductor $224$
Sign $-0.766 + 0.642i$
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  + (−1.53 + 0.884i)3-s + (4.10 − 7.10i)5-s + (−5.46 − 4.37i)7-s + (−2.93 + 5.08i)9-s + (−9.10 + 5.25i)11-s − 5.87·13-s + 14.5i·15-s + (−13.5 − 23.3i)17-s + (10.6 + 6.15i)19-s + (12.2 + 1.85i)21-s + (−36.2 − 20.9i)23-s + (−21.1 − 36.6i)25-s − 26.3i·27-s + 36.4·29-s + (6.61 − 3.82i)31-s + ⋯
L(s)  = 1  + (−0.510 + 0.294i)3-s + (0.820 − 1.42i)5-s + (−0.781 − 0.624i)7-s + (−0.326 + 0.564i)9-s + (−0.827 + 0.477i)11-s − 0.451·13-s + 0.967i·15-s + (−0.794 − 1.37i)17-s + (0.561 + 0.323i)19-s + (0.582 + 0.0885i)21-s + (−1.57 − 0.911i)23-s + (−0.845 − 1.46i)25-s − 0.974i·27-s + 1.25·29-s + (0.213 − 0.123i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.766 + 0.642i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.228094 - 0.626733i\)
\(L(\frac12)\) \(\approx\) \(0.228094 - 0.626733i\)
\(L(2)\) 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 \( 1 + (5.46 + 4.37i)T \)
good3 \( 1 + (1.53 - 0.884i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.10 + 7.10i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (9.10 - 5.25i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 5.87T + 169T^{2} \)
17 \( 1 + (13.5 + 23.3i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-10.6 - 6.15i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (36.2 + 20.9i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 36.4T + 841T^{2} \)
31 \( 1 + (-6.61 + 3.82i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (3.62 - 6.28i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 48.1T + 1.68e3T^{2} \)
43 \( 1 + 23.4iT - 1.84e3T^{2} \)
47 \( 1 + (-51.1 - 29.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-2.03 - 3.52i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-23.2 + 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (0.185 - 0.320i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.80 + 1.04i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 121. iT - 5.04e3T^{2} \)
73 \( 1 + (-38.5 - 66.8i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (31.8 + 18.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 98.2iT - 6.88e3T^{2} \)
89 \( 1 + (-15.1 + 26.2i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 5.87T + 9.40e3T^{2} \)
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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

−11.84339018212612732668807356420, −10.32565082082322259363832817428, −9.935031784345907018588270850289, −8.841149001198438775722680899431, −7.69259097523149692700849323970, −6.29433252227002769622795702633, −5.16133112988875363642427636921, −4.53955084502801741846565358841, −2.39400933499699871607615840961, −0.35661053400609934463062093197, 2.33403517947043855015322225223, 3.40126733596866771551526150816, 5.62704517663276368830614918559, 6.20861701899019029434536806550, 7.00875333521324226628456424373, 8.486208732855720585307933234122, 9.779676082222433620234029553852, 10.40753194552365423542227195635, 11.42585373366331041967095021278, 12.32837981597048203905756974950

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