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Properties

Label 2-56-8.3-c2-0-1
Degree $2$
Conductor $56$
Sign $-0.640 - 0.768i$
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  + (1.37 + 1.45i)2-s − 5.22·3-s + (−0.240 + 3.99i)4-s + 6.26i·5-s + (−7.16 − 7.60i)6-s − 2.64i·7-s + (−6.14 + 5.12i)8-s + 18.2·9-s + (−9.12 + 8.59i)10-s + 9.80·11-s + (1.25 − 20.8i)12-s + 2.41i·13-s + (3.85 − 3.62i)14-s − 32.7i·15-s + (−15.8 − 1.92i)16-s + 6.89·17-s + ⋯
L(s)  = 1  + (0.685 + 0.728i)2-s − 1.74·3-s + (−0.0602 + 0.998i)4-s + 1.25i·5-s + (−1.19 − 1.26i)6-s − 0.377i·7-s + (−0.768 + 0.640i)8-s + 2.03·9-s + (−0.912 + 0.859i)10-s + 0.891·11-s + (0.104 − 1.73i)12-s + 0.185i·13-s + (0.275 − 0.259i)14-s − 2.18i·15-s + (−0.992 − 0.120i)16-s + 0.405·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $-0.640 - 0.768i$
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: \((2,\ 56,\ (\ :1),\ -0.640 - 0.768i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.399558 + 0.853398i\)
\(L(\frac12)\) \(\approx\) \(0.399558 + 0.853398i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 1.45i)T \)
7 \( 1 + 2.64iT \)
good3 \( 1 + 5.22T + 9T^{2} \)
5 \( 1 - 6.26iT - 25T^{2} \)
11 \( 1 - 9.80T + 121T^{2} \)
13 \( 1 - 2.41iT - 169T^{2} \)
17 \( 1 - 6.89T + 289T^{2} \)
19 \( 1 - 2.77T + 361T^{2} \)
23 \( 1 - 42.8iT - 529T^{2} \)
29 \( 1 + 37.3iT - 841T^{2} \)
31 \( 1 + 7.16iT - 961T^{2} \)
37 \( 1 + 0.202iT - 1.36e3T^{2} \)
41 \( 1 - 63.5T + 1.68e3T^{2} \)
43 \( 1 + 35.3T + 1.84e3T^{2} \)
47 \( 1 + 37.9iT - 2.20e3T^{2} \)
53 \( 1 + 54.6iT - 2.80e3T^{2} \)
59 \( 1 - 104.T + 3.48e3T^{2} \)
61 \( 1 - 43.7iT - 3.72e3T^{2} \)
67 \( 1 - 31.1T + 4.48e3T^{2} \)
71 \( 1 - 23.1iT - 5.04e3T^{2} \)
73 \( 1 + 69.2T + 5.32e3T^{2} \)
79 \( 1 - 19.9iT - 6.24e3T^{2} \)
83 \( 1 + 5.11T + 6.88e3T^{2} \)
89 \( 1 + 17.9T + 7.92e3T^{2} \)
97 \( 1 - 12.4T + 9.40e3T^{2} \)
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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

−15.50070602429511540219978942564, −14.42179148995466266843373607262, −13.22696386066631146362103752409, −11.75510782213426122931400982559, −11.33944454593721392301608160266, −9.899921760282082462597569462044, −7.40000410550168923154637215558, −6.57362636089948573256769527920, −5.58079347019406665365074131610, −3.91805967173436858910043170347, 1.01316270294192321577775258844, 4.44557232353856118308418237111, 5.37318725499127924624946832431, 6.48616986582558272898531218663, 9.034657390832059475494784576784, 10.37957236622867979644066132455, 11.47944580775102887898715326159, 12.41553072774258127767723268911, 12.75132173780912633268626085361, 14.49033186939720064318400425704

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