Properties

Label 2-1792-56.11-c0-0-1
Degree $2$
Conductor $1792$
Sign $-0.197 - 0.980i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.866 − 0.5i)5-s + i·7-s + (−0.5 + 0.866i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s − 27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (0.866 − 0.5i)47-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (0.866 − 0.5i)5-s + i·7-s + (−0.5 + 0.866i)11-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.866 − 0.5i)21-s + (0.866 − 0.5i)23-s − 27-s + (0.866 + 0.5i)31-s + (−0.499 − 0.866i)33-s + (0.5 + 0.866i)35-s + (−0.866 + 0.5i)37-s + (0.866 − 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.197 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.197 - 0.980i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1663, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.197 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.023148036\)
\(L(\frac12)\) \(\approx\) \(1.023148036\)
\(L(1)\) 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 \( 1 - iT \)
good3 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.855539640884952660908453764488, −8.883526481227635267557586338812, −8.539733206121923304721844218461, −7.19735059357150107154717962147, −6.30392643777435479660454004803, −5.39721941077021707979494147490, −4.98049159816657324743026059224, −4.18667464773596365279267402766, −2.67374484209718768372937777335, −1.81741518880511112825319749343, 0.826320707957913952098483638792, 2.05427606741120708158007354215, 3.16956897222994540849375548485, 4.27928939984757762591806973471, 5.51459625276506014789135165387, 6.11022700008236262197360786205, 6.89742545502721203061821063778, 7.38747680532847967741619811925, 8.339186966133633638538282795514, 9.345582491746437012333586800738

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