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Properties

Label 2-56-8.3-c2-0-7
Degree $2$
Conductor $56$
Sign $-0.0954 + 0.995i$
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  + (−1.05 + 1.69i)2-s − 3.44·3-s + (−1.77 − 3.58i)4-s − 4.88i·5-s + (3.62 − 5.84i)6-s − 2.64i·7-s + (7.96 + 0.763i)8-s + 2.84·9-s + (8.29 + 5.14i)10-s − 21.4·11-s + (6.10 + 12.3i)12-s − 13.0i·13-s + (4.49 + 2.79i)14-s + 16.8i·15-s + (−9.69 + 12.7i)16-s − 0.234·17-s + ⋯
L(s)  = 1  + (−0.527 + 0.849i)2-s − 1.14·3-s + (−0.443 − 0.896i)4-s − 0.976i·5-s + (0.604 − 0.974i)6-s − 0.377i·7-s + (0.995 + 0.0954i)8-s + 0.315·9-s + (0.829 + 0.514i)10-s − 1.95·11-s + (0.509 + 1.02i)12-s − 1.00i·13-s + (0.321 + 0.199i)14-s + 1.12i·15-s + (−0.606 + 0.795i)16-s − 0.0138·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.0954 + 0.995i$
Analytic conductor: \(1.52588\)
Root analytic conductor: \(1.23526\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1),\ -0.0954 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.211528 - 0.232782i\)
\(L(\frac12)\) \(\approx\) \(0.211528 - 0.232782i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 - 1.69i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 3.44T + 9T^{2} \)
5 \( 1 + 4.88iT - 25T^{2} \)
11 \( 1 + 21.4T + 121T^{2} \)
13 \( 1 + 13.0iT - 169T^{2} \)
17 \( 1 + 0.234T + 289T^{2} \)
19 \( 1 - 4.55T + 361T^{2} \)
23 \( 1 - 10.9iT - 529T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 + 34.1iT - 961T^{2} \)
37 \( 1 + 54.2iT - 1.36e3T^{2} \)
41 \( 1 + 37.8T + 1.68e3T^{2} \)
43 \( 1 + 4.84T + 1.84e3T^{2} \)
47 \( 1 + 72.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.6iT - 2.80e3T^{2} \)
59 \( 1 - 34.9T + 3.48e3T^{2} \)
61 \( 1 - 63.6iT - 3.72e3T^{2} \)
67 \( 1 - 18.4T + 4.48e3T^{2} \)
71 \( 1 + 47.5iT - 5.04e3T^{2} \)
73 \( 1 - 55.9T + 5.32e3T^{2} \)
79 \( 1 + 95.0iT - 6.24e3T^{2} \)
83 \( 1 - 71.5T + 6.88e3T^{2} \)
89 \( 1 + 159.T + 7.92e3T^{2} \)
97 \( 1 + 90.4T + 9.40e3T^{2} \)
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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

−15.20367196106219673576037097281, −13.47451582547582696682224941134, −12.62615706045204264173081583756, −10.94983618011028237571813328318, −10.13365786548915268893483858380, −8.517670852939903065223583235596, −7.40046103141546564497054509480, −5.61069342292341079188917965945, −5.06655906075711888793528675865, −0.40113159036813125091222350800, 2.72928896055146672295760503560, 5.00599829828753315537542256157, 6.70690368386254828997260112102, 8.203975876722358702457372932494, 9.970134196500561489795328847457, 10.82743976303860465335092982325, 11.57941419531719861037663892890, 12.66670340806617549049184863378, 13.94925853656422100644424624496, 15.61069407227110494045701580260

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