Properties

Label 2-1568-7.6-c2-0-52
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

\(L(\frac{3}{2})\) \(\approx\) \(0.7396006767\)
\(L(\frac12)\) \(\approx\) \(0.7396006767\)
\(L(2)\) 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

−8.647558940705532117582377313924, −7.88555214074116528156229533817, −7.06547353172798269315543834780, −6.88885803663832563852975036711, −5.95886186950706053374730559585, −4.95824103047918488894895917347, −3.32240140403060116798414814552, −2.71140400859060549080973286226, −1.85180229487372684623286014777, −0.21097670369069429745310531594, 1.14637478568764462971823678654, 2.69848556593852309494240984072, 3.89129765703054159050525161614, 4.74036175189574036474380048905, 5.12159234091450996508991900308, 5.83688273199572841609454626724, 7.41439016337261621595451178190, 8.222645411497838505842750985850, 8.919920654929177958425571334478, 9.452665369218326490199440077195

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