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Properties

Label 2-1568-7.6-c2-0-31
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  + 0.842i·3-s + 1.40i·5-s + 8.28·9-s − 14.8·11-s − 2.67i·13-s − 1.18·15-s + 13.5i·17-s − 29.1i·19-s + 22.8·23-s + 23.0·25-s + 14.5i·27-s − 3.76·29-s − 13.2i·31-s − 12.5i·33-s + 2.64·37-s + ⋯
L(s)  = 1  + 0.280i·3-s + 0.281i·5-s + 0.921·9-s − 1.35·11-s − 0.205i·13-s − 0.0789·15-s + 0.797i·17-s − 1.53i·19-s + 0.994·23-s + 0.921·25-s + 0.539i·27-s − 0.129·29-s − 0.428i·31-s − 0.380i·33-s + 0.0714·37-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

\(L(\frac{3}{2})\) \(\approx\) \(1.953556989\)
\(L(\frac12)\) \(\approx\) \(1.953556989\)
\(L(2)\) 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 \( 1 - 0.842iT - 9T^{2} \)
5 \( 1 - 1.40iT - 25T^{2} \)
11 \( 1 + 14.8T + 121T^{2} \)
13 \( 1 + 2.67iT - 169T^{2} \)
17 \( 1 - 13.5iT - 289T^{2} \)
19 \( 1 + 29.1iT - 361T^{2} \)
23 \( 1 - 22.8T + 529T^{2} \)
29 \( 1 + 3.76T + 841T^{2} \)
31 \( 1 + 13.2iT - 961T^{2} \)
37 \( 1 - 2.64T + 1.36e3T^{2} \)
41 \( 1 - 45.2iT - 1.68e3T^{2} \)
43 \( 1 - 51.5T + 1.84e3T^{2} \)
47 \( 1 - 67.1iT - 2.20e3T^{2} \)
53 \( 1 - 39.9T + 2.80e3T^{2} \)
59 \( 1 - 12.7iT - 3.48e3T^{2} \)
61 \( 1 + 77.8iT - 3.72e3T^{2} \)
67 \( 1 - 45.8T + 4.48e3T^{2} \)
71 \( 1 + 40.6T + 5.04e3T^{2} \)
73 \( 1 - 64.2iT - 5.32e3T^{2} \)
79 \( 1 + 80.5T + 6.24e3T^{2} \)
83 \( 1 + 121. iT - 6.88e3T^{2} \)
89 \( 1 - 45.9iT - 7.92e3T^{2} \)
97 \( 1 - 134. iT - 9.40e3T^{2} \)
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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

−9.397915960144708979168233366632, −8.581528091354498566323509739497, −7.62086411635742788928698469880, −7.06191365055728346690637344682, −6.09402679873746663848075525183, −5.02324694956857517670172916405, −4.47144732368366104886804582302, −3.21468793570370176435587197202, −2.39866788507997476090451033436, −0.899054034040930692890792492328, 0.70884615625316058728867861242, 1.91154138258348214407654904776, 3.00188843694022213907729009026, 4.15624311224112403710477496440, 5.07040582109407192016401466295, 5.75865623315376302688418354136, 7.02088284513651629446842722249, 7.40566627022243135834373697778, 8.341129736867448590255624833148, 9.084083799817713803122805023492

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