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Properties

Label 2-56-7.2-c1-0-1
Degree $2$
Conductor $56$
Sign $0.386 + 0.922i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
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  + (−1.5 − 2.59i)3-s + (0.5 − 0.866i)5-s + (2 + 1.73i)7-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)11-s + 2·13-s − 3·15-s + (−1.5 − 2.59i)17-s + (−2.5 + 4.33i)19-s + (1.5 − 7.79i)21-s + (1.5 − 2.59i)23-s + (2 + 3.46i)25-s + 9·27-s − 6·29-s + (0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.866 − 1.49i)3-s + (0.223 − 0.387i)5-s + (0.755 + 0.654i)7-s + (−1 + 1.73i)9-s + (0.150 + 0.261i)11-s + 0.554·13-s − 0.774·15-s + (−0.363 − 0.630i)17-s + (−0.573 + 0.993i)19-s + (0.327 − 1.70i)21-s + (0.312 − 0.541i)23-s + (0.400 + 0.692i)25-s + 1.73·27-s − 1.11·29-s + (0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611727 - 0.406909i\)
\(L(\frac12)\) \(\approx\) \(0.611727 - 0.406909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.5 - 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 + 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−14.95036903226213436645933411100, −13.64634290129883345826855146101, −12.69366067660129257240872095062, −11.86672095023638690750641091131, −10.89906967327213489504165293918, −8.859041642410459046960176387808, −7.64440779996722150762856840388, −6.31388810281972817628763161017, −5.17063541996666485357012921658, −1.76846724099755612032672074698, 3.87701761008051489127417062745, 5.11112800413170815637287925945, 6.56227563216336332560750120500, 8.613194565707461501191185933750, 10.00406633894279305696986491913, 10.90686188159101635745940938538, 11.49841620452065568321808967167, 13.39493068958989750796753253955, 14.74678045598644006759510080436, 15.45180652684651068427060106111

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