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Properties

Label 2-56-56.19-c1-0-1
Degree $2$
Conductor $56$
Sign $0.441 - 0.897i$
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  + (1.13 + 0.849i)2-s + (−2.27 + 1.31i)3-s + (0.557 + 1.92i)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 + (2.70 − 1.15i)10-s + (−0.669 − 1.16i)11-s + (−3.78 − 3.63i)12-s − 2.50·13-s + (3.39 − 1.57i)14-s + 5.45i·15-s + (−3.37 + 2.14i)16-s + (−2.78 + 1.60i)17-s + ⋯
L(s)  = 1  + (0.799 + 0.600i)2-s + (−1.31 + 0.757i)3-s + (0.278 + 0.960i)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 + (0.855 − 0.364i)10-s + (−0.201 − 0.349i)11-s + (−1.09 − 1.04i)12-s − 0.694·13-s + (0.907 − 0.420i)14-s + 1.40i·15-s + (−0.844 + 0.535i)16-s + (−0.674 + 0.389i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804836 + 0.500862i\)
\(L(\frac12)\) \(\approx\) \(0.804836 + 0.500862i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 - 0.849i)T \)
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

−15.77228296756215142739867411367, −14.38950722512448974997828922647, −13.27623712788279172751234816440, −12.11796250142579091364993192655, −11.12353991589371337530795556462, −9.872809738609773829353296190313, −8.045567169523044853982536930177, −6.34986348440746678746222414382, −5.17630353464326626934445769754, −4.31549444666244711267873393724, 2.29130353705919315743837983486, 5.05122156712507315950861322546, 6.04184206484354384104801019845, 7.17108296688901489727894066805, 9.712082498989422444435286026439, 10.99533788809695497484851187483, 11.74806605034860180647331489075, 12.56696083983733526556796661872, 13.77189725128595900952760797073, 14.86542586655210524298658002483

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