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Properties

Label 2-224-224.13-c2-0-32
Degree $2$
Conductor $224$
Sign $0.116 + 0.993i$
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  + (−1.99 + 0.125i)2-s + (3.96 − 0.5i)4-s + (4.94 − 4.94i)7-s + (−7.85 + 1.49i)8-s + (−6.36 − 6.36i)9-s + (0.978 + 2.36i)11-s + (−9.26 + 10.5i)14-s + (15.5 − 3.96i)16-s + (13.5 + 11.9i)18-s + (−2.24 − 4.59i)22-s + (−30.1 − 30.1i)23-s + (17.6 − 17.6i)25-s + (17.1 − 22.1i)28-s + (15.9 − 38.6i)29-s + (−30.4 + 9.86i)32-s + ⋯
L(s)  = 1  + (−0.998 + 0.0626i)2-s + (0.992 − 0.125i)4-s + (0.707 − 0.707i)7-s + (−0.982 + 0.186i)8-s + (−0.707 − 0.707i)9-s + (0.0889 + 0.214i)11-s + (−0.661 + 0.750i)14-s + (0.968 − 0.248i)16-s + (0.750 + 0.661i)18-s + (−0.102 − 0.208i)22-s + (−1.31 − 1.31i)23-s + (0.707 − 0.707i)25-s + (0.613 − 0.789i)28-s + (0.551 − 1.33i)29-s + (−0.951 + 0.308i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.116 + 0.993i$
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: \((2,\ 224,\ (\ :1),\ 0.116 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.638373 - 0.567752i\)
\(L(\frac12)\) \(\approx\) \(0.638373 - 0.567752i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.125i)T \)
7 \( 1 + (-4.94 + 4.94i)T \)
good3 \( 1 + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-0.978 - 2.36i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-119. - 119. i)T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + (-255. - 255. i)T^{2} \)
23 \( 1 + (30.1 + 30.1i)T + 529iT^{2} \)
29 \( 1 + (-15.9 + 38.6i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + (-40.7 + 16.8i)T + (968. - 968. i)T^{2} \)
41 \( 1 - 1.68e3iT^{2} \)
43 \( 1 + (30.9 + 74.7i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 + (-38.2 - 92.4i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-5.16 + 12.4i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (59.8 - 59.8i)T - 5.04e3iT^{2} \)
73 \( 1 - 5.32e3iT^{2} \)
79 \( 1 - 156. iT - 6.24e3T^{2} \)
83 \( 1 + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + 7.92e3iT^{2} \)
97 \( 1 - 9.40e3T^{2} \)
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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

−11.67222204129159582963578216913, −10.66385835937602393366329163044, −9.916859030897682465095800774895, −8.709685869909267358722313080615, −8.031497534975760384002597355915, −6.87596435258605758572239923155, −5.89770818450760047660531649615, −4.18264968019418156429204174567, −2.43980599056720149603566143360, −0.64674409885536234715189849728, 1.69139949105388682926758068511, 3.05884279588131679403929938856, 5.13848750672266783615886887979, 6.18528186618418129416882759810, 7.61324880725681765667084455362, 8.343667865937299176829706802724, 9.177513850994962873619542004277, 10.29936585299638100935168839402, 11.35354778560493416265676375338, 11.73939571008204195606787919919

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