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Properties

Label 2-224-32.11-c2-0-29
Degree $2$
Conductor $224$
Sign $-0.0475 + 0.998i$
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  + (−1.86 − 0.710i)2-s + (0.216 + 0.522i)3-s + (2.99 + 2.65i)4-s + (2.53 − 6.12i)5-s + (−0.0336 − 1.13i)6-s + (1.87 − 1.87i)7-s + (−3.70 − 7.08i)8-s + (6.13 − 6.13i)9-s + (−9.09 + 9.65i)10-s + (−7.96 + 19.2i)11-s + (−0.739 + 2.13i)12-s + (−5.18 − 12.5i)13-s + (−4.82 + 2.16i)14-s + 3.74·15-s + (1.89 + 15.8i)16-s − 4.24i·17-s + ⋯
L(s)  = 1  + (−0.934 − 0.355i)2-s + (0.0721 + 0.174i)3-s + (0.747 + 0.663i)4-s + (0.507 − 1.22i)5-s + (−0.00560 − 0.188i)6-s + (0.267 − 0.267i)7-s + (−0.463 − 0.886i)8-s + (0.681 − 0.681i)9-s + (−0.909 + 0.965i)10-s + (−0.724 + 1.74i)11-s + (−0.0616 + 0.178i)12-s + (−0.398 − 0.963i)13-s + (−0.344 + 0.154i)14-s + 0.249·15-s + (0.118 + 0.992i)16-s − 0.249i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0475 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0475 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.0475 + 0.998i$
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: \((2,\ 224,\ (\ :1),\ -0.0475 + 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.752387 - 0.789050i\)
\(L(\frac12)\) \(\approx\) \(0.752387 - 0.789050i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.86 + 0.710i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (-0.216 - 0.522i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-2.53 + 6.12i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (7.96 - 19.2i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (5.18 + 12.5i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 4.24iT - 289T^{2} \)
19 \( 1 + (-30.5 + 12.6i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (26.4 + 26.4i)T + 529iT^{2} \)
29 \( 1 + (18.1 - 7.50i)T + (594. - 594. i)T^{2} \)
31 \( 1 + 23.3iT - 961T^{2} \)
37 \( 1 + (-16.0 + 38.7i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-11.2 + 11.2i)T - 1.68e3iT^{2} \)
43 \( 1 + (-5.38 + 12.9i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 2.38T + 2.20e3T^{2} \)
53 \( 1 + (-35.3 - 14.6i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (9.54 + 3.95i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-16.5 + 6.84i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-41.6 - 100. i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (35.1 - 35.1i)T - 5.04e3iT^{2} \)
73 \( 1 + (90.2 - 90.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 91.9T + 6.24e3T^{2} \)
83 \( 1 + (-76.3 + 31.6i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-18.9 - 18.9i)T + 7.92e3iT^{2} \)
97 \( 1 + 28.9T + 9.40e3T^{2} \)
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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

−11.90483946475029630873582312388, −10.38734000590097384601045144301, −9.763752026919355258120708203386, −9.145958516797511089975896992582, −7.83701194556712124787290849431, −7.14546687958002448011991384238, −5.39026179250386148241088939602, −4.22617812925805706267229548851, −2.30900382191002330757587919078, −0.827411223557050457219000715291, 1.76212951971380785716676175070, 3.08682738217244920250066574211, 5.43379141709653840099430582367, 6.29970711478500416068421785421, 7.45357358196364234868656935035, 8.097595041237888173746146218549, 9.473105663454031831323898280396, 10.25911662234621753269721264069, 11.07819041742020154595409893415, 11.83295352495770180365067112474

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