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Properties

Label 2-224-56.13-c2-0-2
Degree $2$
Conductor $224$
Sign $-0.756 - 0.654i$
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  + 1.75·3-s − 6.54·5-s + (0.267 + 6.99i)7-s − 5.92·9-s + 2.13i·11-s − 12.6·13-s − 11.4·15-s + 24.2i·17-s − 13.8·19-s + (0.469 + 12.2i)21-s + 24.7·23-s + 17.7·25-s − 26.1·27-s − 43.6i·29-s + 34.4i·31-s + ⋯
L(s)  = 1  + 0.584·3-s − 1.30·5-s + (0.0382 + 0.999i)7-s − 0.658·9-s + 0.194i·11-s − 0.970·13-s − 0.764·15-s + 1.42i·17-s − 0.731·19-s + (0.0223 + 0.583i)21-s + 1.07·23-s + 0.711·25-s − 0.969·27-s − 1.50i·29-s + 1.11i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(6.10355\)
Root analytic conductor: \(2.47053\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1),\ -0.756 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.236434 + 0.634298i\)
\(L(\frac12)\) \(\approx\) \(0.236434 + 0.634298i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.267 - 6.99i)T \)
good3 \( 1 - 1.75T + 9T^{2} \)
5 \( 1 + 6.54T + 25T^{2} \)
11 \( 1 - 2.13iT - 121T^{2} \)
13 \( 1 + 12.6T + 169T^{2} \)
17 \( 1 - 24.2iT - 289T^{2} \)
19 \( 1 + 13.8T + 361T^{2} \)
23 \( 1 - 24.7T + 529T^{2} \)
29 \( 1 + 43.6iT - 841T^{2} \)
31 \( 1 - 34.4iT - 961T^{2} \)
37 \( 1 - 27.6iT - 1.36e3T^{2} \)
41 \( 1 + 27.9iT - 1.68e3T^{2} \)
43 \( 1 - 42.6iT - 1.84e3T^{2} \)
47 \( 1 + 34.4iT - 2.20e3T^{2} \)
53 \( 1 - 35.0iT - 2.80e3T^{2} \)
59 \( 1 - 55.7T + 3.48e3T^{2} \)
61 \( 1 + 6.54T + 3.72e3T^{2} \)
67 \( 1 + 42.6iT - 4.48e3T^{2} \)
71 \( 1 + 22.9T + 5.04e3T^{2} \)
73 \( 1 + 59.7iT - 5.32e3T^{2} \)
79 \( 1 - 35.1T + 6.24e3T^{2} \)
83 \( 1 - 144.T + 6.88e3T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 - 136. iT - 9.40e3T^{2} \)
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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

−12.22758045152872955624918576141, −11.65975734601390594196786586945, −10.55866190320732507575567079650, −9.186855113460394007071972736714, −8.383535339665034710926073533588, −7.73094429035685676288633696096, −6.33019880854752068569019729847, −4.92368902692402293099116356651, −3.62689722125866929912387210896, −2.40117289947807833176195373570, 0.32839578363334233012225806498, 2.82930778454771823480907952464, 3.93139982347541672456352138162, 5.07232421101281672237972892294, 6.99280391219939965894080118861, 7.59881111904478780600078284623, 8.567152678380359514421050559131, 9.582736790870992803444473066711, 10.92466377862794113642042253177, 11.51513421933134740682503116178

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