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Properties

Label 2-1792-56.51-c0-0-0
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: $\chi_{1792} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1792,\ (\ :0),\ 0.197 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.582136683\)
\(L(\frac12)\) \(\approx\) \(1.582136683\)
\(L(1)\) 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

−9.590523917125176115393931515323, −9.109376055681951414531095148716, −8.389000001001769751794882072529, −7.05369609965290597974715042040, −6.55593093233507886417818046976, −5.47087354091277510379641971871, −4.76659039149825758193909530565, −3.74316100458043508979310139353, −2.70998939651767025189754511942, −1.99167113684591483642074168646, 1.31782827417036651652519749898, 1.90988638246042662286316963311, 3.35407054574075086358081396378, 4.19493583380334173053446224714, 5.42963537732886919071993501048, 6.16661348590221150116335886339, 6.96636907079564851656186985618, 7.81969120410366231771802232440, 8.366985647391006733664729949450, 9.234229625168497901141763352559

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