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Properties

Label 2-1568-7.6-c2-0-17
Degree $2$
Conductor $1568$
Sign $-0.755 + 0.654i$
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  + 3.98i·3-s + 9.01i·5-s − 6.88·9-s + 16.5·11-s − 0.446i·13-s − 35.9·15-s + 6.95i·17-s + 12.7i·19-s − 26.5·23-s − 56.3·25-s + 8.43i·27-s + 26.4·29-s − 25.1i·31-s + 66.0i·33-s − 63.3·37-s + ⋯
L(s)  = 1  + 1.32i·3-s + 1.80i·5-s − 0.764·9-s + 1.50·11-s − 0.0343i·13-s − 2.39·15-s + 0.408i·17-s + 0.670i·19-s − 1.15·23-s − 2.25·25-s + 0.312i·27-s + 0.912·29-s − 0.810i·31-s + 2.00i·33-s − 1.71·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.755 + 0.654i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ -0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.642099516\)
\(L(\frac12)\) \(\approx\) \(1.642099516\)
\(L(2)\) 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 \( 1 - 3.98iT - 9T^{2} \)
5 \( 1 - 9.01iT - 25T^{2} \)
11 \( 1 - 16.5T + 121T^{2} \)
13 \( 1 + 0.446iT - 169T^{2} \)
17 \( 1 - 6.95iT - 289T^{2} \)
19 \( 1 - 12.7iT - 361T^{2} \)
23 \( 1 + 26.5T + 529T^{2} \)
29 \( 1 - 26.4T + 841T^{2} \)
31 \( 1 + 25.1iT - 961T^{2} \)
37 \( 1 + 63.3T + 1.36e3T^{2} \)
41 \( 1 + 0.519iT - 1.68e3T^{2} \)
43 \( 1 + 25.5T + 1.84e3T^{2} \)
47 \( 1 - 68.6iT - 2.20e3T^{2} \)
53 \( 1 + 7.17T + 2.80e3T^{2} \)
59 \( 1 - 75.4iT - 3.48e3T^{2} \)
61 \( 1 + 46.0iT - 3.72e3T^{2} \)
67 \( 1 - 42.8T + 4.48e3T^{2} \)
71 \( 1 - 60.0T + 5.04e3T^{2} \)
73 \( 1 + 46.7iT - 5.32e3T^{2} \)
79 \( 1 + 54.3T + 6.24e3T^{2} \)
83 \( 1 - 11.4iT - 6.88e3T^{2} \)
89 \( 1 + 61.4iT - 7.92e3T^{2} \)
97 \( 1 + 20.3iT - 9.40e3T^{2} \)
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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

−10.01667236036767635050162508879, −9.195291885887363192219860128096, −8.218274854283796197833391554305, −7.19678950333822453221452155190, −6.41575003769368575260602687722, −5.82052644931894907932700928751, −4.43063389491697844235140814239, −3.74967668537710939351456816855, −3.17719206017776598644269089208, −1.86055250732722048237017097320, 0.46048140732699692132638642269, 1.30280271558532416468374912397, 2.01751488806282163849399197101, 3.69931937307220878125160706645, 4.65805957909142312281248061261, 5.47098429029637745517886952844, 6.51634030665654764963781378881, 7.03101899965389436196145591073, 8.159887756333590750506024920776, 8.591715535620990344194908756731

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