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Properties

Label 2-56-56.5-c2-0-3
Degree $2$
Conductor $56$
Sign $-0.373 - 0.927i$
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  + (−1.70 + 1.04i)2-s + (2.78 + 4.81i)3-s + (1.83 − 3.55i)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 + (−0.147 − 6.11i)10-s + (0.106 − 0.0612i)11-s + (22.2 − 1.07i)12-s + 4.11·13-s + (6.22 + 12.5i)14-s − 17.0·15-s + (−9.30 − 13.0i)16-s + (17.8 − 10.3i)17-s + ⋯
L(s)  = 1  + (−0.853 + 0.520i)2-s + (0.926 + 1.60i)3-s + (0.457 − 0.889i)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 + (−0.0147 − 0.611i)10-s + (0.00963 − 0.00556i)11-s + (1.85 − 0.0895i)12-s + 0.316·13-s + (0.444 + 0.895i)14-s − 1.13·15-s + (−0.581 − 0.813i)16-s + (1.05 − 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.373 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.373 - 0.927i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ -0.373 - 0.927i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.560665 + 0.829776i\)
\(L(\frac12)\) \(\approx\) \(0.560665 + 0.829776i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.70 - 1.04i)T \)
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

−15.47188011105111980030970954028, −14.54521784764634534283273444456, −13.89421377467189535994819317007, −11.16358954501770120023488476367, −10.42772619855433101692863722416, −9.532561646532255996597947381677, −8.370075704685317807962840606423, −7.19431322943286654617070676528, −4.99417566942839463791573283823, −3.30342927293788360566246740036, 1.45685132640652901819058086072, 3.06780053697920432960299180981, 6.35443900825607958584068168304, 7.932560399977474634897350780752, 8.374885264268929823036951319568, 9.562638604191433120587557814208, 11.66250004457020701937519531056, 12.35729688091859612125212822136, 13.13702672075445211872608872809, 14.52846125664644497624755549785

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