Properties

Label 2-224-32.11-c2-0-22
Degree $2$
Conductor $224$
Sign $0.897 - 0.440i$
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  + (−0.518 + 1.93i)2-s + (−0.266 − 0.642i)3-s + (−3.46 − 2.00i)4-s + (−0.747 + 1.80i)5-s + (1.37 − 0.181i)6-s + (1.87 − 1.87i)7-s + (5.66 − 5.65i)8-s + (6.02 − 6.02i)9-s + (−3.09 − 2.37i)10-s + (0.854 − 2.06i)11-s + (−0.364 + 2.75i)12-s + (−2.40 − 5.80i)13-s + (2.64 + 4.58i)14-s + 1.35·15-s + (7.98 + 13.8i)16-s + 18.1i·17-s + ⋯
L(s)  = 1  + (−0.259 + 0.965i)2-s + (−0.0887 − 0.214i)3-s + (−0.865 − 0.500i)4-s + (−0.149 + 0.360i)5-s + (0.229 − 0.0301i)6-s + (0.267 − 0.267i)7-s + (0.707 − 0.706i)8-s + (0.669 − 0.669i)9-s + (−0.309 − 0.237i)10-s + (0.0776 − 0.187i)11-s + (−0.0304 + 0.229i)12-s + (−0.184 − 0.446i)13-s + (0.188 + 0.327i)14-s + 0.0905·15-s + (0.498 + 0.866i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ 0.897 - 0.440i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.25188 + 0.290804i\)
\(L(\frac12)\) \(\approx\) \(1.25188 + 0.290804i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.518 - 1.93i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (0.266 + 0.642i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (0.747 - 1.80i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-0.854 + 2.06i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (2.40 + 5.80i)T + (-119. + 119. i)T^{2} \)
17 \( 1 - 18.1iT - 289T^{2} \)
19 \( 1 + (-34.1 + 14.1i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (-6.06 - 6.06i)T + 529iT^{2} \)
29 \( 1 + (-29.1 + 12.0i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 17.4iT - 961T^{2} \)
37 \( 1 + (-5.11 + 12.3i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (19.9 - 19.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17.0 + 41.2i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 35.7T + 2.20e3T^{2} \)
53 \( 1 + (87.6 + 36.3i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-62.7 - 25.9i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (91.3 - 37.8i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (28.7 + 69.5i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (-47.0 + 47.0i)T - 5.04e3iT^{2} \)
73 \( 1 + (-25.5 + 25.5i)T - 5.32e3iT^{2} \)
79 \( 1 + 8.38T + 6.24e3T^{2} \)
83 \( 1 + (12.8 - 5.32i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-72.9 - 72.9i)T + 7.92e3iT^{2} \)
97 \( 1 + 102.T + 9.40e3T^{2} \)
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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

−12.22185735954056438257281850330, −10.97206741784719100027025297795, −9.967925315075757121380321925450, −9.080329059877003391054992415650, −7.83257802356778945195859734295, −7.13481484012584314739085137866, −6.14274962366680251979635461229, −4.91826691602908430121488834726, −3.55789154506581713526781698867, −1.00688543648893899848217759562, 1.30821076265378061159246483821, 2.91118767690948242228664881376, 4.48058060753797638968655545974, 5.17718583936339818827702061213, 7.20039212841341334429368795556, 8.181215112201475966279562195880, 9.294130292068811493009498138103, 10.01432494170156953240789224213, 11.02181547201896161977709027833, 11.96352195675211200493877688530

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