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Properties

Label 2-1568-8.3-c2-0-54
Degree $2$
Conductor $1568$
Sign $0.943 + 0.330i$
Analytic cond. $42.7249$
Root an. cond. $6.53642$
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  + 1.64·3-s + 4.56i·5-s − 6.28·9-s + 12.3·11-s − 18.3i·13-s + 7.52i·15-s + 13.0·17-s − 3.02·19-s − 30.3i·23-s + 4.19·25-s − 25.1·27-s − 22.7i·29-s + 22.5i·31-s + 20.4·33-s − 13.7i·37-s + ⋯
L(s)  = 1  + 0.549·3-s + 0.912i·5-s − 0.697·9-s + 1.12·11-s − 1.41i·13-s + 0.501i·15-s + 0.766·17-s − 0.159·19-s − 1.31i·23-s + 0.167·25-s − 0.933·27-s − 0.785i·29-s + 0.727i·31-s + 0.618·33-s − 0.372i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.943 + 0.330i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.943 + 0.330i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.401070989\)
\(L(\frac12)\) \(\approx\) \(2.401070989\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 1.64T + 9T^{2} \)
5 \( 1 - 4.56iT - 25T^{2} \)
11 \( 1 - 12.3T + 121T^{2} \)
13 \( 1 + 18.3iT - 169T^{2} \)
17 \( 1 - 13.0T + 289T^{2} \)
19 \( 1 + 3.02T + 361T^{2} \)
23 \( 1 + 30.3iT - 529T^{2} \)
29 \( 1 + 22.7iT - 841T^{2} \)
31 \( 1 - 22.5iT - 961T^{2} \)
37 \( 1 + 13.7iT - 1.36e3T^{2} \)
41 \( 1 - 60.5T + 1.68e3T^{2} \)
43 \( 1 + 39.0T + 1.84e3T^{2} \)
47 \( 1 - 20.3iT - 2.20e3T^{2} \)
53 \( 1 - 4.76iT - 2.80e3T^{2} \)
59 \( 1 - 11.7T + 3.48e3T^{2} \)
61 \( 1 + 108. iT - 3.72e3T^{2} \)
67 \( 1 - 79.1T + 4.48e3T^{2} \)
71 \( 1 + 12.9iT - 5.04e3T^{2} \)
73 \( 1 - 98.5T + 5.32e3T^{2} \)
79 \( 1 - 131. iT - 6.24e3T^{2} \)
83 \( 1 - 28.3T + 6.88e3T^{2} \)
89 \( 1 - 157.T + 7.92e3T^{2} \)
97 \( 1 - 39.6T + 9.40e3T^{2} \)
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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

−9.186556903851716335896178061808, −8.298069733199221231438357591996, −7.78508996312280007258107078500, −6.71167485076860506654229770235, −6.10998034747752027775933123944, −5.12642080763741420366907043736, −3.81394019550361506455689599206, −3.11483756763658451078048963641, −2.33358371217836724220002132183, −0.71727620885967321015820546475, 1.06954913281221397320710555831, 2.03693785261531938397701576922, 3.39024991748378892324776435557, 4.13226466579646377069057945924, 5.11862319701990124629051943772, 6.00203575875134030722867538344, 6.92118682996372697715574109648, 7.84499120901490947828509422824, 8.680466818991431539321081549156, 9.222067363511894531462207441877

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