Properties

Label 2-1568-56.13-c2-0-41
Degree $2$
Conductor $1568$
Sign $0.987 - 0.160i$
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  − 3.89·3-s + 8.85·5-s + 6.18·9-s − 3.64i·11-s − 7.79·13-s − 34.5·15-s + 10.4i·17-s + 10.7·19-s + 12.9·23-s + 53.4·25-s + 10.9·27-s + 17.2i·29-s − 30.2i·31-s + 14.2i·33-s + 39.5i·37-s + ⋯
L(s)  = 1  − 1.29·3-s + 1.77·5-s + 0.686·9-s − 0.331i·11-s − 0.599·13-s − 2.30·15-s + 0.616i·17-s + 0.567·19-s + 0.561·23-s + 2.13·25-s + 0.406·27-s + 0.594i·29-s − 0.975i·31-s + 0.430i·33-s + 1.06i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.987 - 0.160i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.710225287\)
\(L(\frac12)\) \(\approx\) \(1.710225287\)
\(L(2)\) 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 \( 1 + 3.89T + 9T^{2} \)
5 \( 1 - 8.85T + 25T^{2} \)
11 \( 1 + 3.64iT - 121T^{2} \)
13 \( 1 + 7.79T + 169T^{2} \)
17 \( 1 - 10.4iT - 289T^{2} \)
19 \( 1 - 10.7T + 361T^{2} \)
23 \( 1 - 12.9T + 529T^{2} \)
29 \( 1 - 17.2iT - 841T^{2} \)
31 \( 1 + 30.2iT - 961T^{2} \)
37 \( 1 - 39.5iT - 1.36e3T^{2} \)
41 \( 1 + 73.6iT - 1.68e3T^{2} \)
43 \( 1 - 40.8iT - 1.84e3T^{2} \)
47 \( 1 - 41.8iT - 2.20e3T^{2} \)
53 \( 1 - 6.41iT - 2.80e3T^{2} \)
59 \( 1 + 15.9T + 3.48e3T^{2} \)
61 \( 1 - 12.1T + 3.72e3T^{2} \)
67 \( 1 - 7.79iT - 4.48e3T^{2} \)
71 \( 1 - 41.3T + 5.04e3T^{2} \)
73 \( 1 + 89.6iT - 5.32e3T^{2} \)
79 \( 1 + 70.7T + 6.24e3T^{2} \)
83 \( 1 - 60.8T + 6.88e3T^{2} \)
89 \( 1 + 27.0iT - 7.92e3T^{2} \)
97 \( 1 - 3.26iT - 9.40e3T^{2} \)
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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

−9.467458693384469223865360884291, −8.689461426831939192134523612439, −7.41345569637786755744046513677, −6.48206018327061248888302779055, −5.96984531068307720057511167336, −5.34053119912415108258282530148, −4.66662435448856250915634258783, −3.06360434136942807216396789383, −1.92618605548121408009007987716, −0.847944439779846908601984398279, 0.76010118436500014015642837476, 1.91585845838343473366189104178, 2.94722771683891204571960221722, 4.66442475180012533846728452856, 5.28530433298807928439290829648, 5.81327903440790366414351424224, 6.66252916718926267940459709494, 7.22838163938605732328171245088, 8.629860500077863321380425544486, 9.566939130616614490870692881383

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