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Properties

Label 2-224-28.3-c1-0-6
Degree $2$
Conductor $224$
Sign $0.531 + 0.847i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.13 − 1.96i)3-s + (3.08 − 1.77i)5-s + (−0.912 + 2.48i)7-s + (−1.06 − 1.85i)9-s + (−2.80 − 1.61i)11-s + 5.65i·13-s − 8.06i·15-s + (−3.39 − 1.96i)17-s + (−0.220 − 0.382i)19-s + (3.84 + 4.60i)21-s + (−0.781 + 0.451i)23-s + (3.82 − 6.63i)25-s + 1.95·27-s + 2.13·29-s + (−1.32 + 2.30i)31-s + ⋯
L(s)  = 1  + (0.654 − 1.13i)3-s + (1.37 − 0.795i)5-s + (−0.345 + 0.938i)7-s + (−0.356 − 0.617i)9-s + (−0.844 − 0.487i)11-s + 1.56i·13-s − 2.08i·15-s + (−0.824 − 0.475i)17-s + (−0.0506 − 0.0876i)19-s + (0.838 + 1.00i)21-s + (−0.162 + 0.0940i)23-s + (0.765 − 1.32i)25-s + 0.375·27-s + 0.397·29-s + (−0.238 + 0.413i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.531 + 0.847i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.531 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45016 - 0.802513i\)
\(L(\frac12)\) \(\approx\) \(1.45016 - 0.802513i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.912 - 2.48i)T \)
good3 \( 1 + (-1.13 + 1.96i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-3.08 + 1.77i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.80 + 1.61i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.65iT - 13T^{2} \)
17 \( 1 + (3.39 + 1.96i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.220 + 0.382i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.781 - 0.451i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.13T + 29T^{2} \)
31 \( 1 + (1.32 - 2.30i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.01 + 5.21i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 6.19iT - 43T^{2} \)
47 \( 1 + (0.496 + 0.860i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.183 - 0.317i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.78 + 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.28 - 3.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.53 - 4.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.12iT - 71T^{2} \)
73 \( 1 + (3.19 + 1.84i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.01 - 3.47i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 + (5.13 - 2.96i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.43iT - 97T^{2} \)
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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

−12.50658549274532848873794622649, −11.34844428965754752969442472459, −9.796210725836559780275041689531, −8.991816796126717284578494740317, −8.393037577470129340079447249962, −6.92404538810087661272639750205, −6.05927958780894674445190855854, −4.92475545967745347951391329800, −2.60774089315212297117789578710, −1.77705942258397122974698148354, 2.49958791633125547357453336382, 3.57607190850639744995318460947, 4.98519266612370009917424974407, 6.19477974416061714028268950262, 7.41135681347590845061198956612, 8.702571687179430794485385423989, 9.915623863324500810263364402070, 10.27898378092525401770175489736, 10.75508928883398877098155851540, 12.76986416711507391734487100189

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