Properties

Label 2-56-56.51-c2-0-12
Degree $2$
Conductor $56$
Sign $-0.685 + 0.727i$
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  + (0.371 − 1.96i)2-s + (−0.824 − 1.42i)3-s + (−3.72 − 1.46i)4-s + (−3.95 − 2.28i)5-s + (−3.11 + 1.08i)6-s + (6.75 − 1.83i)7-s + (−4.25 + 6.77i)8-s + (3.14 − 5.43i)9-s + (−5.94 + 6.91i)10-s + (6.18 + 10.7i)11-s + (0.984 + 6.52i)12-s − 18.3i·13-s + (−1.09 − 13.9i)14-s + 7.52i·15-s + (11.7 + 10.8i)16-s + (6.51 + 11.2i)17-s + ⋯
L(s)  = 1  + (0.185 − 0.982i)2-s + (−0.274 − 0.475i)3-s + (−0.930 − 0.365i)4-s + (−0.790 − 0.456i)5-s + (−0.518 + 0.181i)6-s + (0.965 − 0.262i)7-s + (−0.531 + 0.846i)8-s + (0.348 − 0.604i)9-s + (−0.594 + 0.691i)10-s + (0.562 + 0.974i)11-s + (0.0820 + 0.543i)12-s − 1.41i·13-s + (−0.0781 − 0.996i)14-s + 0.501i·15-s + (0.733 + 0.679i)16-s + (0.383 + 0.663i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.685 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.685 + 0.727i$
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: \((2,\ 56,\ (\ :1),\ -0.685 + 0.727i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.411502 - 0.953107i\)
\(L(\frac12)\) \(\approx\) \(0.411502 - 0.953107i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.371 + 1.96i)T \)
7 \( 1 + (-6.75 + 1.83i)T \)
good3 \( 1 + (0.824 + 1.42i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (3.95 + 2.28i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-6.18 - 10.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 18.3iT - 169T^{2} \)
17 \( 1 + (-6.51 - 11.2i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-1.51 + 2.61i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-26.2 - 15.1i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 22.7iT - 841T^{2} \)
31 \( 1 + (19.5 - 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (11.9 + 6.88i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 60.5T + 1.68e3T^{2} \)
43 \( 1 - 39.0T + 1.84e3T^{2} \)
47 \( 1 + (-17.6 - 10.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (4.12 - 2.38i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (5.86 + 10.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-94.3 - 54.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.5 - 68.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 12.9iT - 5.04e3T^{2} \)
73 \( 1 + (-49.2 - 85.3i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (113. + 65.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 28.3T + 6.88e3T^{2} \)
89 \( 1 + (-78.7 + 136. i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 39.6T + 9.40e3T^{2} \)
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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

−14.58189062904345778076753222063, −12.96784976841467915982156280535, −12.33224475346118032002716503148, −11.41760278457539290397932505566, −10.19751282130778174111260211516, −8.690096885278269235248409747796, −7.40250781005276545626327316511, −5.23021135031226284728355255787, −3.83701454749898579404060146152, −1.20785803032827447992187209538, 3.98296609570170217316784601899, 5.19755129657755294766367444275, 6.88512408694693146557903079950, 8.054195855715536496068856208566, 9.266539681281050907884900016950, 11.03787551317793626733635973101, 11.86052568645371411197760157331, 13.65744097656428729145813379866, 14.50532193872743237238008141936, 15.46450775355818286071502769130

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