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Properties

Label 2-56-8.3-c2-0-8
Degree $2$
Conductor $56$
Sign $0.906 + 0.421i$
Analytic cond. $1.52588$
Root an. cond. $1.23526$
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  + (1.85 − 0.739i)2-s + 0.0974·3-s + (2.90 − 2.74i)4-s + 3.46i·5-s + (0.181 − 0.0720i)6-s − 2.64i·7-s + (3.37 − 7.25i)8-s − 8.99·9-s + (2.56 + 6.44i)10-s − 2.92·11-s + (0.283 − 0.267i)12-s + 19.1i·13-s + (−1.95 − 4.91i)14-s + 0.337i·15-s + (0.902 − 15.9i)16-s − 14.3·17-s + ⋯
L(s)  = 1  + (0.929 − 0.369i)2-s + 0.0324·3-s + (0.726 − 0.686i)4-s + 0.693i·5-s + (0.0301 − 0.0120i)6-s − 0.377i·7-s + (0.421 − 0.906i)8-s − 0.998·9-s + (0.256 + 0.644i)10-s − 0.266·11-s + (0.0236 − 0.0223i)12-s + 1.47i·13-s + (−0.139 − 0.351i)14-s + 0.0225i·15-s + (0.0563 − 0.998i)16-s − 0.846·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.906 + 0.421i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.75692 - 0.388291i\)
\(L(\frac12)\) \(\approx\) \(1.75692 - 0.388291i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.85 + 0.739i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 - 0.0974T + 9T^{2} \)
5 \( 1 - 3.46iT - 25T^{2} \)
11 \( 1 + 2.92T + 121T^{2} \)
13 \( 1 - 19.1iT - 169T^{2} \)
17 \( 1 + 14.3T + 289T^{2} \)
19 \( 1 - 8.09T + 361T^{2} \)
23 \( 1 + 16.7iT - 529T^{2} \)
29 \( 1 - 27.1iT - 841T^{2} \)
31 \( 1 + 44.8iT - 961T^{2} \)
37 \( 1 + 39.5iT - 1.36e3T^{2} \)
41 \( 1 - 45.8T + 1.68e3T^{2} \)
43 \( 1 - 61.0T + 1.84e3T^{2} \)
47 \( 1 - 46.2iT - 2.20e3T^{2} \)
53 \( 1 + 9.69iT - 2.80e3T^{2} \)
59 \( 1 + 114.T + 3.48e3T^{2} \)
61 \( 1 - 7.48iT - 3.72e3T^{2} \)
67 \( 1 + 12.0T + 4.48e3T^{2} \)
71 \( 1 - 129. iT - 5.04e3T^{2} \)
73 \( 1 + 18.2T + 5.32e3T^{2} \)
79 \( 1 - 42.6iT - 6.24e3T^{2} \)
83 \( 1 + 109.T + 6.88e3T^{2} \)
89 \( 1 + 80.9T + 7.92e3T^{2} \)
97 \( 1 - 162.T + 9.40e3T^{2} \)
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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

−14.42736614144911594964672435569, −14.12026684446456303835629179843, −12.75958568472193757741482127682, −11.39197685337973939566182683134, −10.82058136496622199755791285843, −9.219770440177292508059258463590, −7.21460805783126498343122320424, −6.04951841765891415464662996156, −4.31043738524039274457625280890, −2.60670644644854021553731466612, 2.97495578888134730478562963596, 4.96527729488269925598314760827, 5.97136455438983833901146784333, 7.78201367128832737972224905961, 8.845862371007607084640703064208, 10.80370527904314320821800305570, 12.00674786433998944507712825076, 12.94386615435318225632214483198, 13.91158121469127602282494036507, 15.16430213402536024171547492841

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