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Properties

Label 2-56-56.45-c2-0-0
Degree $2$
Conductor $56$
Sign $0.235 - 0.971i$
Analytic cond. $1.52588$
Root an. cond. $1.23526$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0483 − 1.99i)2-s + (−2.78 + 4.81i)3-s + (−3.99 + 0.193i)4-s + (1.52 + 2.64i)5-s + (9.76 + 5.32i)6-s + (0.608 + 6.97i)7-s + (0.579 + 7.97i)8-s + (−10.9 − 18.9i)9-s + (5.22 − 3.18i)10-s + (−0.106 − 0.0612i)11-s + (10.1 − 19.7i)12-s − 4.11·13-s + (13.9 − 1.55i)14-s − 17.0·15-s + (15.9 − 1.54i)16-s + (17.8 + 10.3i)17-s + ⋯
L(s)  = 1  + (−0.0241 − 0.999i)2-s + (−0.926 + 1.60i)3-s + (−0.998 + 0.0483i)4-s + (0.305 + 0.529i)5-s + (1.62 + 0.887i)6-s + (0.0868 + 0.996i)7-s + (0.0724 + 0.997i)8-s + (−1.21 − 2.10i)9-s + (0.522 − 0.318i)10-s + (−0.00963 − 0.00556i)11-s + (0.848 − 1.64i)12-s − 0.316·13-s + (0.993 − 0.110i)14-s − 1.13·15-s + (0.995 − 0.0965i)16-s + (1.05 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.583302 + 0.459028i\)
\(L(\frac12)\) \(\approx\) \(0.583302 + 0.459028i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0483 + 1.99i)T \)
7 \( 1 + (-0.608 - 6.97i)T \)
good3 \( 1 + (2.78 - 4.81i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.52 - 2.64i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (0.106 + 0.0612i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 4.11T + 169T^{2} \)
17 \( 1 + (-17.8 - 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (4.46 + 7.74i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.51 - 13.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 31.6iT - 841T^{2} \)
31 \( 1 + (-23.0 - 13.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-25.1 + 14.5i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 9.26iT - 1.68e3T^{2} \)
43 \( 1 + 45.3iT - 1.84e3T^{2} \)
47 \( 1 + (68.6 - 39.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-55.0 - 31.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-14.2 + 24.6i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-12.6 - 21.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (65.4 + 37.8i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 2.81T + 5.04e3T^{2} \)
73 \( 1 + (-11.0 - 6.40i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (35.6 + 61.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 30.0T + 6.88e3T^{2} \)
89 \( 1 + (-15.3 + 8.83i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 26.1iT - 9.40e3T^{2} \)
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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

−15.13996706445511315641966097906, −14.41290714933562996725079971703, −12.49390048668520311878980068412, −11.57763645965381573073736202009, −10.60781509470938590242017810308, −9.828472333357616260328845909633, −8.786943746738343918853510936067, −5.87968072920996615510190380211, −4.80761489849397567537443359937, −3.17020784891202856981640906515, 0.880058615048759538810205449076, 4.96728106629185923511064339085, 6.20631055740434367251085693208, 7.28326955158747699992256826914, 8.147632912892011270774130645506, 10.02835752179748890762120528262, 11.68739946204376515558843030220, 12.90052715868774518446877632738, 13.47894742678303505117817235470, 14.53309000550267483866913599600

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