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Properties

Label 2-1792-56.51-c0-0-3
Degree $2$
Conductor $1792$
Sign $-0.980 - 0.197i$
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

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

Invariants

Degree: \(2\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.980 - 0.197i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ -0.980 - 0.197i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2805743832\)
\(L(\frac12)\) \(\approx\) \(0.2805743832\)
\(L(1)\) 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

−8.849820468524537487070727735223, −8.161231243049619037377703541598, −7.63190447264036119939575093191, −6.51850298501953270022830576239, −5.97189343762250297436720480127, −5.09712246972117181513083375882, −4.09100826103085448346625305264, −2.95914926478444190105262568794, −1.75506005225363068231710062209, −0.21465879143512795629350936273, 2.00943977961081110898327009254, 3.47624011676942146643317409881, 4.23103050687062354108591294440, 4.67657075539469320318203498803, 5.81258249262142013388132142431, 6.84673841201784765932089243989, 7.54595475639170080642710349477, 8.089498539036542899285108457135, 9.374499619947417920513617672338, 10.01101147544704990243433308913

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