Properties

Label 2-56-56.51-c2-0-13
Degree $2$
Conductor $56$
Sign $-0.966 - 0.256i$
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  + (−1.19 − 1.60i)2-s + (−2.66 − 4.61i)3-s + (−1.13 + 3.83i)4-s + (1.86 + 1.07i)5-s + (−4.21 + 9.79i)6-s + (−6.91 − 1.06i)7-s + (7.50 − 2.76i)8-s + (−9.71 + 16.8i)9-s + (−0.506 − 4.28i)10-s + (−2.62 − 4.55i)11-s + (20.7 − 4.97i)12-s − 21.4i·13-s + (6.57 + 12.3i)14-s − 11.5i·15-s + (−13.4 − 8.71i)16-s + (−0.463 − 0.802i)17-s + ⋯
L(s)  = 1  + (−0.598 − 0.801i)2-s + (−0.888 − 1.53i)3-s + (−0.284 + 0.958i)4-s + (0.373 + 0.215i)5-s + (−0.701 + 1.63i)6-s + (−0.988 − 0.152i)7-s + (0.938 − 0.345i)8-s + (−1.07 + 1.86i)9-s + (−0.0506 − 0.428i)10-s + (−0.239 − 0.414i)11-s + (1.72 − 0.414i)12-s − 1.64i·13-s + (0.469 + 0.882i)14-s − 0.766i·15-s + (−0.838 − 0.544i)16-s + (−0.0272 − 0.0472i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.966 - 0.256i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ -0.966 - 0.256i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0605751 + 0.464002i\)
\(L(\frac12)\) \(\approx\) \(0.0605751 + 0.464002i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 + 1.60i)T \)
7 \( 1 + (6.91 + 1.06i)T \)
good3 \( 1 + (2.66 + 4.61i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.86 - 1.07i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.62 + 4.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + (0.463 + 0.802i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-2.96 + 5.13i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.52 - 4.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 9.42iT - 841T^{2} \)
31 \( 1 + (-29.8 + 17.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (11.0 + 6.40i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 43.1T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + (-39.8 - 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-64.5 + 37.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (26.8 + 46.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (24.0 + 13.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 - 67.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (16.8 + 29.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.1 + 15.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 72.9T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 + 47.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 53.7T + 9.40e3T^{2} \)
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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

−13.63798351559569442129480042766, −13.01584359389768775474974410864, −12.22846238708737450351188349693, −11.02245052086254833658422747347, −10.03072664695394181000582613322, −8.219489269082987389317618063680, −7.08169332353483734518737060694, −5.78203724640360005591674670601, −2.75689121037447894498636506594, −0.60810229487898187397247251842, 4.33705001874117224469151027255, 5.62592919008448001696166003827, 6.73032678110954977339795531956, 9.039209000192181869908007681457, 9.654376591482618104117822901281, 10.56684313107544147646086255484, 11.90770023661133184854105504975, 13.73482933335553119689137149413, 15.07315018904771617394816187402, 15.85984923675904131613515803504

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