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Properties

Label 2-56-56.27-c1-0-3
Degree $2$
Conductor $56$
Sign $0.921 + 0.387i$
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 + i)2-s − 2.44i·3-s − 2i·4-s + 2.44·5-s + (2.44 + 2.44i)6-s + (−2.44 + i)7-s + (2 + 2i)8-s − 2.99·9-s + (−2.44 + 2.44i)10-s + 2·11-s − 4.89·12-s − 2.44·13-s + (1.44 − 3.44i)14-s − 5.99i·15-s − 4·16-s + 4.89i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.41i·3-s − i·4-s + 1.09·5-s + (0.999 + 0.999i)6-s + (−0.925 + 0.377i)7-s + (0.707 + 0.707i)8-s − 0.999·9-s + (−0.774 + 0.774i)10-s + 0.603·11-s − 1.41·12-s − 0.679·13-s + (0.387 − 0.921i)14-s − 1.54i·15-s − 16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.921 + 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663048 - 0.133647i\)
\(L(\frac12)\) \(\approx\) \(0.663048 - 0.133647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
7 \( 1 + (2.44 - i)T \)
good3 \( 1 + 2.44iT - 3T^{2} \)
5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 2.44T + 13T^{2} \)
17 \( 1 - 4.89iT - 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 2.44iT - 59T^{2} \)
61 \( 1 + 7.34T + 61T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 + 14.6iT - 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 + 2.44iT - 83T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−15.19909128324404508440629829222, −13.99854981557860414600959468955, −13.17888468152953520332413681531, −12.04582241795520484651057415173, −10.16690116372717326804670333419, −9.203458247507367328718954086447, −7.79291351632568902133719206267, −6.50144943066760156297205247485, −5.88953540509910196551585205691, −1.89198746771187162038117391700, 3.02359056856572027054513349814, 4.69890912599202400122281514675, 6.77502625487572142392429167873, 8.944406995191295677098499948995, 9.809691152486119517803241295809, 10.19320327450609072983773264087, 11.59234796763125983740622182117, 13.05187881440594372745787453448, 14.18016073769462053272801153769, 15.71139716301795103182580458437

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