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Properties

Label 2-224-56.13-c2-0-10
Degree $2$
Conductor $224$
Sign $0.929 + 0.369i$
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  + 4.11·3-s − 1.10·5-s + (3.73 − 5.92i)7-s + 7.92·9-s − 10.7i·11-s + 13.1·13-s − 4.53·15-s + 20.5i·17-s + 24.3·19-s + (15.3 − 24.3i)21-s − 16.7·23-s − 23.7·25-s − 4.40·27-s − 4.21i·29-s + 52.8i·31-s + ⋯
L(s)  = 1  + 1.37·3-s − 0.220·5-s + (0.533 − 0.846i)7-s + 0.880·9-s − 0.976i·11-s + 1.01·13-s − 0.302·15-s + 1.20i·17-s + 1.28·19-s + (0.731 − 1.16i)21-s − 0.729·23-s − 0.951·25-s − 0.163·27-s − 0.145i·29-s + 1.70i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.929 + 0.369i$
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: \((2,\ 224,\ (\ :1),\ 0.929 + 0.369i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.39618 - 0.458684i\)
\(L(\frac12)\) \(\approx\) \(2.39618 - 0.458684i\)
\(L(2)\) 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 \( 1 + (-3.73 + 5.92i)T \)
good3 \( 1 - 4.11T + 9T^{2} \)
5 \( 1 + 1.10T + 25T^{2} \)
11 \( 1 + 10.7iT - 121T^{2} \)
13 \( 1 - 13.1T + 169T^{2} \)
17 \( 1 - 20.5iT - 289T^{2} \)
19 \( 1 - 24.3T + 361T^{2} \)
23 \( 1 + 16.7T + 529T^{2} \)
29 \( 1 + 4.21iT - 841T^{2} \)
31 \( 1 - 52.8iT - 961T^{2} \)
37 \( 1 - 9.97iT - 1.36e3T^{2} \)
41 \( 1 - 23.6iT - 1.68e3T^{2} \)
43 \( 1 + 65.2iT - 1.84e3T^{2} \)
47 \( 1 + 52.8iT - 2.20e3T^{2} \)
53 \( 1 - 47.1iT - 2.80e3T^{2} \)
59 \( 1 + 40.2T + 3.48e3T^{2} \)
61 \( 1 + 1.10T + 3.72e3T^{2} \)
67 \( 1 - 65.2iT - 4.48e3T^{2} \)
71 \( 1 + 113.T + 5.04e3T^{2} \)
73 \( 1 - 91.5iT - 5.32e3T^{2} \)
79 \( 1 + 27.1T + 6.24e3T^{2} \)
83 \( 1 + 2.34T + 6.88e3T^{2} \)
89 \( 1 - 50.5iT - 7.92e3T^{2} \)
97 \( 1 + 74.2iT - 9.40e3T^{2} \)
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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

−11.93252635955198184577054062158, −10.90090161674125506675092206767, −9.993138482563489761609988287499, −8.646756899908982527751246836197, −8.248367146842832109041399914007, −7.24977355827253307740456672820, −5.76800000397888735993802400349, −4.00157874572068700404005186493, −3.29993502019693684336048668249, −1.48169356081608833380705374446, 1.92206822351628093500876189768, 3.10802690722814979061282755122, 4.43107070145227568687145553909, 5.84938424905875303060060309651, 7.52365062781380228694818291381, 8.042189197400169246828852655550, 9.226278201011565912173553261864, 9.665122023583246183652497920808, 11.31594839899063358777905228091, 12.02944896319210647374452196653

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