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Properties

Label 2-56-56.51-c2-0-0
Degree $2$
Conductor $56$
Sign $-0.126 - 0.991i$
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  + (−1.78 + 0.907i)2-s + (1.99 + 3.44i)3-s + (2.35 − 3.23i)4-s + (1.63 + 0.941i)5-s + (−6.67 − 4.33i)6-s + (−5.14 + 4.74i)7-s + (−1.25 + 7.90i)8-s + (−3.42 + 5.93i)9-s + (−3.75 − 0.197i)10-s + (3.93 + 6.82i)11-s + (15.8 + 1.66i)12-s − 11.4i·13-s + (4.86 − 13.1i)14-s + 7.49i·15-s + (−4.94 − 15.2i)16-s + (1.44 + 2.51i)17-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (0.663 + 1.14i)3-s + (0.587 − 0.808i)4-s + (0.326 + 0.188i)5-s + (−1.11 − 0.722i)6-s + (−0.735 + 0.677i)7-s + (−0.156 + 0.987i)8-s + (−0.380 + 0.659i)9-s + (−0.375 − 0.0197i)10-s + (0.358 + 0.620i)11-s + (1.31 + 0.138i)12-s − 0.883i·13-s + (0.347 − 0.937i)14-s + 0.499i·15-s + (−0.308 − 0.951i)16-s + (0.0852 + 0.147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.126 - 0.991i$
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: \((2,\ 56,\ (\ :1),\ -0.126 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.634755 + 0.721041i\)
\(L(\frac12)\) \(\approx\) \(0.634755 + 0.721041i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.78 - 0.907i)T \)
7 \( 1 + (5.14 - 4.74i)T \)
good3 \( 1 + (-1.99 - 3.44i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.63 - 0.941i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-3.93 - 6.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 11.4iT - 169T^{2} \)
17 \( 1 + (-1.44 - 2.51i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-15.0 + 26.0i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-33.3 - 19.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 27.8iT - 841T^{2} \)
31 \( 1 + (19.4 - 11.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-39.4 - 22.7i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 40.6T + 1.68e3T^{2} \)
43 \( 1 + 47.2T + 1.84e3T^{2} \)
47 \( 1 + (71.5 + 41.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-23.2 + 13.4i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-5.20 - 9.01i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-19.1 - 11.0i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.6 - 51.3i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 38.2iT - 5.04e3T^{2} \)
73 \( 1 + (6.98 + 12.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (44.3 + 25.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 89.4T + 6.88e3T^{2} \)
89 \( 1 + (52.6 - 91.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 55.3T + 9.40e3T^{2} \)
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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

−15.27270234805243106180303777263, −14.92885607266949005692308574974, −13.33704978896979860357851803881, −11.51720616567082835217830098499, −10.02155323997009505650320985718, −9.569912106895075763917918886862, −8.518867122537763870734787408755, −6.84343150869216779427390230744, −5.23354640524496461040633959110, −2.93851436053456585800980236110, 1.41646101587517734418309292904, 3.30814579716555963817586604544, 6.55847499115392445671138300719, 7.50224410983979878707584641833, 8.762260509214860182282352811194, 9.792878446168873536919265333349, 11.26496959484151970750754348072, 12.58351935761721610483851434949, 13.33710921587187480438771599295, 14.40024765821080708589846141931

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