↑

Properties

Label 1-1792-1792.149-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.996 - 0.0878i$
Analytic cond. $8.32201$
Root an. cond. $8.32201$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.935 + 0.352i)3-s + (0.683 − 0.729i)5-s + (0.751 − 0.659i)9-s + (−0.812 − 0.582i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.528 + 0.849i)19-s + (−0.997 − 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.0327 − 0.999i)37-s + ⋯
L(s)  = 1  + (−0.935 + 0.352i)3-s + (0.683 − 0.729i)5-s + (0.751 − 0.659i)9-s + (−0.812 − 0.582i)11-s + (−0.956 − 0.290i)13-s + (−0.382 + 0.923i)15-s + (0.991 + 0.130i)17-s + (−0.528 + 0.849i)19-s + (−0.997 − 0.0654i)23-s + (−0.0654 − 0.997i)25-s + (−0.471 + 0.881i)27-s + (0.995 − 0.0980i)29-s + (0.965 − 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.0327 − 0.999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.996 - 0.0878i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.996 - 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01240496812 - 0.2820093833i\)
\(L(\frac12)\) \(\approx\) \(0.01240496812 - 0.2820093833i\)
\(L(1)\) \(\approx\) \(0.6892433939 - 0.1038607623i\)
\(L(1)\) \(\approx\) \(0.6892433939 - 0.1038607623i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.935 + 0.352i)T \)
5 \( 1 + (0.683 - 0.729i)T \)
11 \( 1 + (-0.812 - 0.582i)T \)
13 \( 1 + (-0.956 - 0.290i)T \)
17 \( 1 + (0.991 + 0.130i)T \)
19 \( 1 + (-0.528 + 0.849i)T \)
23 \( 1 + (-0.997 - 0.0654i)T \)
29 \( 1 + (0.995 - 0.0980i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (-0.0327 - 0.999i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (-0.773 + 0.634i)T \)
47 \( 1 + (-0.793 - 0.608i)T \)
53 \( 1 + (-0.582 + 0.812i)T \)
59 \( 1 + (0.227 - 0.973i)T \)
61 \( 1 + (-0.986 - 0.162i)T \)
67 \( 1 + (-0.352 - 0.935i)T \)
71 \( 1 + (-0.195 + 0.980i)T \)
73 \( 1 + (-0.321 + 0.946i)T \)
79 \( 1 + (0.130 + 0.991i)T \)
83 \( 1 + (0.881 - 0.471i)T \)
89 \( 1 + (0.442 - 0.896i)T \)
97 \( 1 + (0.707 + 0.707i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.77423456710587905546437258859, −19.56115079787816717794723347776, −18.99543249067497988504885421873, −18.13102821472434696747130583516, −17.743225428000599742514674103249, −17.05249273178301638479271866994, −16.2966681078049655688622029221, −15.374410119926032694460116923630, −14.687677758367247217641019554395, −13.70922614008193155858735782766, −13.19853457222610171972109554179, −12.14034918951874907941586546361, −11.79887495856882107616406622619, −10.62856862344190585207150704539, −10.16260404463924170399939603809, −9.63511042868229720019545036939, −8.196112887493183132595505935755, −7.41542438001569174071956515233, −6.692227480580526094967996553392, −6.0976189317553599112982505794, −5.04892329166293015506966623793, −4.65783024770364062776544287619, −3.097504326624916951974679583623, −2.30258164642355318338196724860, −1.43841212725752431418470869027, 0.11549323647829651905775025745, 1.23639618882722974268583594255, 2.29816687375271249055910802172, 3.47314045644308200135149519989, 4.55390103113912309430386072036, 5.195308403052024034565957932263, 5.84998146678901415095566444047, 6.48729147356975925693327617674, 7.77525059443032309112475632933, 8.37155761061088087497409563015, 9.56686348079065548679120307513, 10.13828345840135534759987459119, 10.5659940239961257355112035081, 11.82340064706245883534755444134, 12.308261111070994689461649023195, 12.9525512191973111994737733729, 13.88170743950455269021603936731, 14.66442669169371552853501012206, 15.70259910614049963006748516825, 16.263430268568714753779598104462, 16.95589309503032279129625547046, 17.44517148792846542468809204501, 18.263402618217358787306073758618, 18.94032701269590258047868799486, 19.98554871956987612056748820055

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