Properties

Label 2-224-8.5-c1-0-2
Degree $2$
Conductor $224$
Sign $0.905 - 0.424i$
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  − 0.936i·3-s + 3.33i·5-s + 7-s + 2.12·9-s + 4.27i·11-s − 3.33i·13-s + 3.12·15-s + 2·17-s − 0.936i·19-s − 0.936i·21-s + 3.12·23-s − 6.12·25-s − 4.79i·27-s − 1.87i·29-s − 6.24·31-s + ⋯
L(s)  = 1  − 0.540i·3-s + 1.49i·5-s + 0.377·7-s + 0.707·9-s + 1.28i·11-s − 0.924i·13-s + 0.806·15-s + 0.485·17-s − 0.214i·19-s − 0.204i·21-s + 0.651·23-s − 1.22·25-s − 0.923i·27-s − 0.347i·29-s − 1.12·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.905 - 0.424i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.905 - 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27618 + 0.283970i\)
\(L(\frac12)\) \(\approx\) \(1.27618 + 0.283970i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 + 0.936iT - 3T^{2} \)
5 \( 1 - 3.33iT - 5T^{2} \)
11 \( 1 - 4.27iT - 11T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 0.936iT - 19T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 1.87iT - 29T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 - 1.87iT - 37T^{2} \)
41 \( 1 + 12.2T + 41T^{2} \)
43 \( 1 - 4.27iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 8.54iT - 53T^{2} \)
59 \( 1 + 7.60iT - 59T^{2} \)
61 \( 1 + 3.33iT - 61T^{2} \)
67 \( 1 + 15.7iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 9.47iT - 83T^{2} \)
89 \( 1 - 0.246T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−12.39039906459207917902234504372, −11.30621786416284103075021084271, −10.38048286852167911973342829444, −9.711772249668447394344779832936, −7.975565808397175921310096753890, −7.22715632871555151422167238199, −6.53894971324182652644403818233, −4.99683523001505722643861053315, −3.39907397124431242017305299152, −1.98277242281230573402704323001, 1.36393720080517626578335742800, 3.72924650051081452777375767085, 4.77508113317440899125380030023, 5.66068158236485121803569069640, 7.25857870779372428760683722162, 8.610602100665214182799760174926, 9.034482040763009840187007145919, 10.18412109204969958804887701260, 11.27269946673338102420603771211, 12.19585775275808448895316825889

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