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Properties

Label 2-224-7.6-c4-0-6
Degree $2$
Conductor $224$
Sign $-0.745 - 0.666i$
Analytic cond. $23.1548$
Root an. cond. $4.81195$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.32i·3-s + 44.6i·5-s + (−36.5 − 32.6i)7-s + 75.6·9-s + 180.·11-s + 167. i·13-s − 103.·15-s − 13.9i·17-s + 400. i·19-s + (75.8 − 84.8i)21-s − 199.·23-s − 1.36e3·25-s + 363. i·27-s − 1.28e3·29-s − 669. i·31-s + ⋯
L(s)  = 1  + 0.258i·3-s + 1.78i·5-s + (−0.745 − 0.666i)7-s + 0.933·9-s + 1.49·11-s + 0.989i·13-s − 0.460·15-s − 0.0482i·17-s + 1.11i·19-s + (0.172 − 0.192i)21-s − 0.376·23-s − 2.18·25-s + 0.498i·27-s − 1.52·29-s − 0.696i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(23.1548\)
Root analytic conductor: \(4.81195\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :2),\ -0.745 - 0.666i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.574259355\)
\(L(\frac12)\) \(\approx\) \(1.574259355\)
\(L(3)\) 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 \( 1 + (36.5 + 32.6i)T \)
good3 \( 1 - 2.32iT - 81T^{2} \)
5 \( 1 - 44.6iT - 625T^{2} \)
11 \( 1 - 180.T + 1.46e4T^{2} \)
13 \( 1 - 167. iT - 2.85e4T^{2} \)
17 \( 1 + 13.9iT - 8.35e4T^{2} \)
19 \( 1 - 400. iT - 1.30e5T^{2} \)
23 \( 1 + 199.T + 2.79e5T^{2} \)
29 \( 1 + 1.28e3T + 7.07e5T^{2} \)
31 \( 1 + 669. iT - 9.23e5T^{2} \)
37 \( 1 - 214.T + 1.87e6T^{2} \)
41 \( 1 - 1.28e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.02e3T + 3.41e6T^{2} \)
47 \( 1 - 1.16e3iT - 4.87e6T^{2} \)
53 \( 1 + 2.38e3T + 7.89e6T^{2} \)
59 \( 1 + 6.80e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.59e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.52e3T + 2.01e7T^{2} \)
71 \( 1 - 3.45e3T + 2.54e7T^{2} \)
73 \( 1 + 178. iT - 2.83e7T^{2} \)
79 \( 1 + 5.60e3T + 3.89e7T^{2} \)
83 \( 1 - 1.16e4iT - 4.74e7T^{2} \)
89 \( 1 + 1.12e4iT - 6.27e7T^{2} \)
97 \( 1 - 1.17e4iT - 8.85e7T^{2} \)
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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

−11.69468602556754851611153985152, −10.97680265224766635576886260162, −9.883487681304740009956727019461, −9.547788303543366558750081780953, −7.63955013437144922767940785862, −6.76160018396118120207377305264, −6.28071996703975295487057502998, −4.04551753711327892483252905311, −3.53426657535323650073166525040, −1.77077820345178922790350757125, 0.54869405763826737765640125604, 1.71714021734058413596465846323, 3.71829338596810417658298761222, 4.88517642744026674504151084530, 5.95309776380202639616569916979, 7.15770050550768870341097953865, 8.497330537298177471699059650021, 9.182955524816285289303448227216, 9.900741615152037832415752413351, 11.55202656126144172100700280791

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