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Properties

Label 2-56-56.53-c3-0-19
Degree $2$
Conductor $56$
Sign $-0.0777 + 0.996i$
Analytic cond. $3.30410$
Root an. cond. $1.81772$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.33 − 2.49i)2-s + (0.982 + 0.567i)3-s + (−4.41 − 6.67i)4-s + (5.55 − 3.20i)5-s + (2.72 − 1.68i)6-s + (10.9 − 14.9i)7-s + (−22.5 + 2.06i)8-s + (−12.8 − 22.2i)9-s + (−0.552 − 18.1i)10-s + (17.8 + 10.2i)11-s + (−0.552 − 9.05i)12-s + 87.1i·13-s + (−22.7 − 47.2i)14-s + 7.27·15-s + (−25.0 + 58.9i)16-s + (19.3 − 33.5i)17-s + ⋯
L(s)  = 1  + (0.473 − 0.880i)2-s + (0.189 + 0.109i)3-s + (−0.551 − 0.833i)4-s + (0.497 − 0.286i)5-s + (0.185 − 0.114i)6-s + (0.588 − 0.808i)7-s + (−0.995 + 0.0912i)8-s + (−0.476 − 0.824i)9-s + (−0.0174 − 0.573i)10-s + (0.488 + 0.282i)11-s + (−0.0132 − 0.217i)12-s + 1.85i·13-s + (−0.433 − 0.901i)14-s + 0.125·15-s + (−0.391 + 0.920i)16-s + (0.276 − 0.478i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0777 + 0.996i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0777 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.0777 + 0.996i$
Analytic conductor: \(3.30410\)
Root analytic conductor: \(1.81772\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :3/2),\ -0.0777 + 0.996i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.27044 - 1.37338i\)
\(L(\frac12)\) \(\approx\) \(1.27044 - 1.37338i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 2.49i)T \)
7 \( 1 + (-10.9 + 14.9i)T \)
good3 \( 1 + (-0.982 - 0.567i)T + (13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.55 + 3.20i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-17.8 - 10.2i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 87.1iT - 2.19e3T^{2} \)
17 \( 1 + (-19.3 + 33.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-55.0 + 31.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-75.7 - 131. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 159. iT - 2.43e4T^{2} \)
31 \( 1 + (-90.2 + 156. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (159. - 92.2i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 320.T + 6.89e4T^{2} \)
43 \( 1 + 272. iT - 7.95e4T^{2} \)
47 \( 1 + (-22.6 - 39.2i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (190. + 109. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-42.0 - 24.2i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-282. + 163. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (619. + 357. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 19.2T + 3.57e5T^{2} \)
73 \( 1 + (-200. + 347. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-238. - 412. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 1.39e3iT - 5.71e5T^{2} \)
89 \( 1 + (33.8 + 58.5i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.43e3T + 9.12e5T^{2} \)
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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

−14.06617556323399569035844647532, −13.60717510512532395736389848026, −11.95027736305652223109152083830, −11.31927124524905691197030321667, −9.693020925493337062786377152090, −9.036615897811666624928683698940, −6.83361423775554416454519953023, −5.08108654916841223421899770241, −3.68310060138994650565467552467, −1.48923065067166201814213629638, 2.89401826715598976172387196651, 5.13918072001839736940977624471, 6.09335165340047687460778747839, 7.87524933167243355088058035168, 8.623692134401350788988835514202, 10.36872381331580617751947303815, 11.93558078546843264132532181359, 13.08622589215318133620428732687, 14.16397937110046943863035854850, 14.88496476627082054467710504406

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