Properties

Label 2-56-56.3-c1-0-3
Degree $2$
Conductor $56$
Sign $-0.262 + 0.965i$
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  + (−1.30 + 0.554i)2-s + (−2.27 − 1.31i)3-s + (1.38 − 1.44i)4-s + (−1.03 − 1.80i)5-s + (3.68 + 0.445i)6-s + (−1.25 − 2.33i)7-s + (−1.00 + 2.64i)8-s + (1.94 + 3.36i)9-s + (2.35 + 1.76i)10-s + (−0.669 + 1.16i)11-s + (−5.03 + 1.46i)12-s + 2.50·13-s + (2.92 + 2.33i)14-s + 5.45i·15-s + (−0.167 − 3.99i)16-s + (−2.78 − 1.60i)17-s + ⋯
L(s)  = 1  + (−0.919 + 0.392i)2-s + (−1.31 − 0.757i)3-s + (0.692 − 0.721i)4-s + (−0.464 − 0.805i)5-s + (1.50 + 0.182i)6-s + (−0.473 − 0.880i)7-s + (−0.353 + 0.935i)8-s + (0.647 + 1.12i)9-s + (0.743 + 0.558i)10-s + (−0.201 + 0.349i)11-s + (−1.45 + 0.422i)12-s + 0.694·13-s + (0.780 + 0.624i)14-s + 1.40i·15-s + (−0.0417 − 0.999i)16-s + (−0.674 − 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.262 + 0.965i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ -0.262 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194471 - 0.254352i\)
\(L(\frac12)\) \(\approx\) \(0.194471 - 0.254352i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.30 - 0.554i)T \)
7 \( 1 + (1.25 + 2.33i)T \)
good3 \( 1 + (2.27 + 1.31i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.03 + 1.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.669 - 1.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.50T + 13T^{2} \)
17 \( 1 + (2.78 + 1.60i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.55 + 2.05i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.54 + 3.20i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.66iT - 29T^{2} \)
31 \( 1 + (2.21 - 3.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.50 - 3.17i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.55iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.565 - 0.980i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.43 - 4.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.29 - 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 + 4.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.93 + 6.81i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 + (-0.480 - 0.277i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.26 - 3.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.503iT - 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−15.59205740217615832091182990484, −13.62648809806179652475167566357, −12.47806456043739083069990631028, −11.39857378400608208412305192215, −10.46918904430642041327905902684, −8.897996461209725388727256319445, −7.39564530922170966069507942056, −6.55309914111481374036790479352, −5.02882862742591161467975464875, −0.76605765556851903447854019152, 3.42421322678768620288794940431, 5.71214577969151355896233847414, 6.98952262271931846056654122838, 8.797805390280997839288269405728, 10.06267593289410585660724041327, 11.11899567974934597711082892133, 11.55117098214183344152960976241, 12.87234800628279114947298458687, 15.15783712661815294446066713823, 15.87335584335559215700615491728

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