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Properties

Label 2-224-56.45-c2-0-8
Degree $2$
Conductor $224$
Sign $0.654 + 0.756i$
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  + (−2.78 + 4.81i)3-s + (−1.52 − 2.64i)5-s + (−0.608 − 6.97i)7-s + (−10.9 − 18.9i)9-s + (−0.106 − 0.0612i)11-s + 4.11·13-s + 17.0·15-s + (17.8 + 10.3i)17-s + (−4.46 − 7.74i)19-s + (35.2 + 16.4i)21-s + (−7.51 − 13.0i)23-s + (7.82 − 13.5i)25-s + 71.8·27-s − 31.6i·29-s + (−23.0 − 13.2i)31-s + ⋯
L(s)  = 1  + (−0.926 + 1.60i)3-s + (−0.305 − 0.529i)5-s + (−0.0868 − 0.996i)7-s + (−1.21 − 2.10i)9-s + (−0.00963 − 0.00556i)11-s + 0.316·13-s + 1.13·15-s + (1.05 + 0.606i)17-s + (−0.235 − 0.407i)19-s + (1.67 + 0.783i)21-s + (−0.326 − 0.566i)23-s + (0.312 − 0.541i)25-s + 2.65·27-s − 1.09i·29-s + (−0.742 − 0.428i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.654 + 0.756i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.654555 - 0.299249i\)
\(L(\frac12)\) \(\approx\) \(0.654555 - 0.299249i\)
\(L(2)\) 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 \( 1 + (0.608 + 6.97i)T \)
good3 \( 1 + (2.78 - 4.81i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.52 + 2.64i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (0.106 + 0.0612i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 - 4.11T + 169T^{2} \)
17 \( 1 + (-17.8 - 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.46 + 7.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (7.51 + 13.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 31.6iT - 841T^{2} \)
31 \( 1 + (23.0 + 13.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (25.1 - 14.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 9.26iT - 1.68e3T^{2} \)
43 \( 1 + 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (-68.6 + 39.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (55.0 + 31.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-14.2 + 24.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (12.6 + 21.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (65.4 + 37.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 2.81T + 5.04e3T^{2} \)
73 \( 1 + (-11.0 - 6.40i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-35.6 - 61.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 30.0T + 6.88e3T^{2} \)
89 \( 1 + (-15.3 + 8.83i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 26.1iT - 9.40e3T^{2} \)
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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

−11.70833579257618042638648072325, −10.69352474914638472228048862355, −10.22317414209934382414408158441, −9.226310140018891109235987646035, −8.090361496770534893869086020831, −6.50895210982796931040843074521, −5.39617413372742055895425109662, −4.35337634650535208631645821393, −3.64797322894021452582885952004, −0.46612297442672134060979550935, 1.50176122273856486509819863595, 3.00328865444154747236481550221, 5.32072764062885642130964283318, 6.03323961367717028289219528487, 7.11455202527574772574350211837, 7.81043900422583740437619139432, 9.033771547424477803642889695185, 10.63823551227880168238001077334, 11.44511303393631366336610901969, 12.23858695538294400765340122621

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