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Properties

Label 1-1792-1792.101-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.0857 + 0.996i$
Analytic cond. $192.577$
Root an. cond. $192.577$
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  + (0.849 + 0.528i)3-s + (0.986 − 0.162i)5-s + (0.442 + 0.896i)9-s + (−0.729 − 0.683i)11-s + (0.773 + 0.634i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (0.910 + 0.412i)19-s + (−0.321 + 0.946i)23-s + (0.946 − 0.321i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.582 − 0.812i)37-s + ⋯
L(s)  = 1  + (0.849 + 0.528i)3-s + (0.986 − 0.162i)5-s + (0.442 + 0.896i)9-s + (−0.729 − 0.683i)11-s + (0.773 + 0.634i)13-s + (0.923 + 0.382i)15-s + (0.793 + 0.608i)17-s + (0.910 + 0.412i)19-s + (−0.321 + 0.946i)23-s + (0.946 − 0.321i)25-s + (−0.0980 + 0.995i)27-s + (−0.290 + 0.956i)29-s + (−0.258 + 0.965i)31-s + (−0.258 − 0.965i)33-s + (0.582 − 0.812i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0857 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0857 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.0857 + 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.0857 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.757420591 + 3.004903791i\)
\(L(\frac12)\) \(\approx\) \(2.757420591 + 3.004903791i\)
\(L(1)\) \(\approx\) \(1.691229980 + 0.5688695197i\)
\(L(1)\) \(\approx\) \(1.691229980 + 0.5688695197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.849 + 0.528i)T \)
5 \( 1 + (0.986 - 0.162i)T \)
11 \( 1 + (-0.729 - 0.683i)T \)
13 \( 1 + (0.773 + 0.634i)T \)
17 \( 1 + (0.793 + 0.608i)T \)
19 \( 1 + (0.910 + 0.412i)T \)
23 \( 1 + (-0.321 + 0.946i)T \)
29 \( 1 + (-0.290 + 0.956i)T \)
31 \( 1 + (-0.258 + 0.965i)T \)
37 \( 1 + (0.582 - 0.812i)T \)
41 \( 1 + (0.195 + 0.980i)T \)
43 \( 1 + (-0.881 - 0.471i)T \)
47 \( 1 + (0.991 + 0.130i)T \)
53 \( 1 + (-0.683 + 0.729i)T \)
59 \( 1 + (-0.935 - 0.352i)T \)
61 \( 1 + (-0.999 - 0.0327i)T \)
67 \( 1 + (0.528 - 0.849i)T \)
71 \( 1 + (-0.555 - 0.831i)T \)
73 \( 1 + (0.0654 - 0.997i)T \)
79 \( 1 + (-0.608 - 0.793i)T \)
83 \( 1 + (0.995 - 0.0980i)T \)
89 \( 1 + (0.659 - 0.751i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.11440975168971626861002081988, −18.73370819514952298688072687381, −18.507963309571016739578677124101, −17.848197597784462226993853585182, −17.068273376492770992174689492635, −16.001274547092496155261455465555, −15.26343865157522718546674942550, −14.54783857910096887189754960577, −13.749319947151553347010804296628, −13.2906220278254257973080668252, −12.62586739971045675489342205198, −11.72723323037867848291835878839, −10.606917511401434383628934800369, −9.82150000599216571332732477982, −9.38595350559037286037363584805, −8.314461040582762484251170042586, −7.68369353584852606898246116853, −6.89141318749687146597394644423, −5.99097048499346722615283669706, −5.26298861408417478859372992073, −4.1195283090819895254197553507, −2.96094595136212489582142906717, −2.51835360071039301743734620973, −1.519192135955467654808054142696, −0.60042236499407819521270983376, 1.27278263054040868123880363287, 1.86688264492554344641006806962, 3.10643708299162054047457227207, 3.51806981125652549577206487026, 4.7545087982725666940676218919, 5.52676528044150150520402584880, 6.19947536426343607730454648938, 7.472836607469180822605487289505, 8.122731141142429120100269533786, 9.08316572748450440156308160829, 9.4345526168107440359688926388, 10.48079251003142847813836118996, 10.82710072000467450239581291420, 12.1019161224722347443316418147, 13.05706287990125650216317739617, 13.701399220529519029231274892416, 14.14187483737906047682107272144, 14.90888216726439862363958190868, 16.01329602035278597316221484109, 16.28636179374110081281860335476, 17.15135186613076609866562663357, 18.34312540091221293710182542526, 18.54444469840973597460839879361, 19.61288301826951596411832825528, 20.34030984349742028807366471465

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