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Properties

Label 2-224-28.3-c1-0-5
Degree $2$
Conductor $224$
Sign $0.719 + 0.694i$
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  + (0.376 − 0.652i)3-s + (1.00 − 0.579i)5-s + (0.286 − 2.63i)7-s + (1.21 + 2.10i)9-s + (−1.47 − 0.850i)11-s − 3.51i·13-s − 0.873i·15-s + (4.54 + 2.62i)17-s + (−0.663 − 1.14i)19-s + (−1.60 − 1.17i)21-s + (1.84 − 1.06i)23-s + (−1.82 + 3.16i)25-s + 4.09·27-s − 2.43·29-s + (−4.27 + 7.39i)31-s + ⋯
L(s)  = 1  + (0.217 − 0.376i)3-s + (0.448 − 0.259i)5-s + (0.108 − 0.994i)7-s + (0.405 + 0.702i)9-s + (−0.444 − 0.256i)11-s − 0.974i·13-s − 0.225i·15-s + (1.10 + 0.636i)17-s + (−0.152 − 0.263i)19-s + (−0.351 − 0.257i)21-s + (0.385 − 0.222i)23-s + (−0.365 + 0.633i)25-s + 0.787·27-s − 0.451·29-s + (−0.767 + 1.32i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.719 + 0.694i$
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: \((2,\ 224,\ (\ :1/2),\ 0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30513 - 0.526718i\)
\(L(\frac12)\) \(\approx\) \(1.30513 - 0.526718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.286 + 2.63i)T \)
good3 \( 1 + (-0.376 + 0.652i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.00 + 0.579i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.47 + 0.850i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.51iT - 13T^{2} \)
17 \( 1 + (-4.54 - 2.62i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.663 + 1.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.84 + 1.06i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.43T + 29T^{2} \)
31 \( 1 + (4.27 - 7.39i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.21 + 5.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.51iT - 41T^{2} \)
43 \( 1 - 12.2iT - 43T^{2} \)
47 \( 1 + (-4.84 - 8.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.04 - 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.13 - 5.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.64 + 1.52i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.4 + 6.60i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (2.10 + 1.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.10 - 2.37i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (-6.59 + 3.80i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−12.50938273566414351463141233528, −10.77125386726521262682050689013, −10.49519440552309277940775495584, −9.204234406120303048491761823420, −7.87924761114489335580630969426, −7.41399671583603215819627440985, −5.89009632212968280863117021621, −4.79544303907159451017345607674, −3.23301658607839894074913508239, −1.43652920794733142189106940677, 2.14692895661492707912197435512, 3.61396042996141590873808918360, 5.07593881647818973890189256196, 6.15851027774396316246546831085, 7.31771047656144058111358807418, 8.655374570835357054872007612366, 9.539664800187458364014961211020, 10.15726610835585586636911946731, 11.58415902853673582250074066225, 12.22115733337878510486473236840

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