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Properties

Label 2-56-56.37-c1-0-1
Degree $2$
Conductor $56$
Sign $0.962 - 0.270i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.114i)2-s + (−0.591 + 0.341i)3-s + (1.97 + 0.323i)4-s + (2.80 + 1.61i)5-s + (0.872 − 0.413i)6-s + (1.47 − 2.19i)7-s + (−2.74 − 0.682i)8-s + (−1.26 + 2.19i)9-s + (−3.76 − 2.60i)10-s + (−2.08 + 1.20i)11-s + (−1.27 + 0.482i)12-s − 3.09i·13-s + (−2.33 + 2.92i)14-s − 2.21·15-s + (3.79 + 1.27i)16-s + (−1.97 − 3.42i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0811i)2-s + (−0.341 + 0.197i)3-s + (0.986 + 0.161i)4-s + (1.25 + 0.724i)5-s + (0.356 − 0.168i)6-s + (0.558 − 0.829i)7-s + (−0.970 − 0.241i)8-s + (−0.422 + 0.731i)9-s + (−1.19 − 0.823i)10-s + (−0.629 + 0.363i)11-s + (−0.368 + 0.139i)12-s − 0.858i·13-s + (−0.624 + 0.781i)14-s − 0.570·15-s + (0.947 + 0.319i)16-s + (−0.479 − 0.830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.611834 + 0.0844103i\)
\(L(\frac12)\) \(\approx\) \(0.611834 + 0.0844103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.114i)T \)
7 \( 1 + (-1.47 + 2.19i)T \)
good3 \( 1 + (0.591 - 0.341i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.80 - 1.61i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 + (1.97 + 3.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.33 + 1.35i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.37 - 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (-1.10 - 1.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.30 + 2.48i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.11T + 41T^{2} \)
43 \( 1 - 11.5iT - 43T^{2} \)
47 \( 1 + (-3.31 + 5.74i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.23 - 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.6 + 6.13i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.54 - 4.35i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.01 - 2.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 + (-4.77 - 8.27i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.838 - 1.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.47iT - 83T^{2} \)
89 \( 1 + (6.98 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.37T + 97T^{2} \)
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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

−15.55578447154864819189550335661, −14.25251983643547380788524962837, −13.19370740772628048372413321069, −11.31449475283696151307981987549, −10.53574938682763732186842670613, −9.841370240116647881099657007637, −8.133853564271474174775724217968, −6.89412630493581947890674673305, −5.40246501335624620235710868533, −2.41995605140237068695672227107, 1.98943746563924340842159249736, 5.52137465019387513554749887144, 6.43165254191398264358571800626, 8.483677744659340434324347382347, 9.108491272708213792274685697445, 10.44519995750804405605908736559, 11.73062504600322105389583325660, 12.73081440555178770977936604133, 14.27102942143770519352199642214, 15.44763097694702474512916399738

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