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Properties

Label 2-224-56.27-c1-0-3
Degree $2$
Conductor $224$
Sign $0.935 + 0.353i$
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  − 2.64i·7-s + 3·9-s + 4·11-s + 5.29i·23-s − 5·25-s − 10.5i·29-s + 10.5i·37-s − 12·43-s − 7.00·49-s + 10.5i·53-s − 7.93i·63-s − 4·67-s − 5.29i·71-s − 10.5i·77-s + 15.8i·79-s + ⋯
L(s)  = 1  − 0.999i·7-s + 9-s + 1.20·11-s + 1.10i·23-s − 25-s − 1.96i·29-s + 1.73i·37-s − 1.82·43-s − 49-s + 1.45i·53-s − 0.999i·63-s − 0.488·67-s − 0.627i·71-s − 1.20i·77-s + 1.78i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(224\)    =    \(2^{5} \cdot 7\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{224} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 224,\ (\ :1/2),\ 0.935 + 0.353i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30065 - 0.237597i\)
\(L(\frac12)\) \(\approx\) \(1.30065 - 0.237597i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 + 10.5iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.06557483860527034299620531922, −11.34436108247883167850609453958, −10.04871488055362295682351441334, −9.582358494613024752970500930116, −8.083173450993270659605003738941, −7.15168892393948979255100184924, −6.21582404203390083723612488505, −4.53270465427190292181413969951, −3.67547683248527082149287337927, −1.45605785642116846050891671960, 1.83878120835424374972243439361, 3.62797910414392303563635931634, 4.92680883399711132158191577086, 6.23831579269273872917902894561, 7.16150935108467611770633917749, 8.546395830076278245343632819159, 9.312480829385573256702764495218, 10.31051710828699044062156525461, 11.50064929263654063357137787404, 12.32180775967608868365646478578

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