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Properties

Label 1-1792-1792.1245-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0245 - 0.999i$
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  + (−0.995 − 0.0980i)3-s + (−0.471 − 0.881i)5-s + (0.980 + 0.195i)9-s + (−0.773 − 0.634i)11-s + (−0.881 − 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (−0.956 + 0.290i)19-s + (−0.831 + 0.555i)23-s + (−0.555 + 0.831i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (−0.290 + 0.956i)37-s + ⋯
L(s)  = 1  + (−0.995 − 0.0980i)3-s + (−0.471 − 0.881i)5-s + (0.980 + 0.195i)9-s + (−0.773 − 0.634i)11-s + (−0.881 − 0.471i)13-s + (0.382 + 0.923i)15-s + (−0.382 + 0.923i)17-s + (−0.956 + 0.290i)19-s + (−0.831 + 0.555i)23-s + (−0.555 + 0.831i)25-s + (−0.956 − 0.290i)27-s + (0.634 + 0.773i)29-s + (0.707 − 0.707i)31-s + (0.707 + 0.707i)33-s + (−0.290 + 0.956i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0245 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0245 - 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0245 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2692032111 - 0.2758928825i\)
\(L(\frac12)\) \(\approx\) \(0.2692032111 - 0.2758928825i\)
\(L(1)\) \(\approx\) \(0.5429796699 - 0.08252855874i\)
\(L(1)\) \(\approx\) \(0.5429796699 - 0.08252855874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.995 - 0.0980i)T \)
5 \( 1 + (-0.471 - 0.881i)T \)
11 \( 1 + (-0.773 - 0.634i)T \)
13 \( 1 + (-0.881 - 0.471i)T \)
17 \( 1 + (-0.382 + 0.923i)T \)
19 \( 1 + (-0.956 + 0.290i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (0.634 + 0.773i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (-0.290 + 0.956i)T \)
41 \( 1 + (0.555 + 0.831i)T \)
43 \( 1 + (-0.995 + 0.0980i)T \)
47 \( 1 + (0.923 + 0.382i)T \)
53 \( 1 + (0.634 - 0.773i)T \)
59 \( 1 + (-0.881 + 0.471i)T \)
61 \( 1 + (0.0980 - 0.995i)T \)
67 \( 1 + (0.0980 - 0.995i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.195 - 0.980i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.290 + 0.956i)T \)
89 \( 1 + (0.831 + 0.555i)T \)
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

−20.16678199725280157218338737717, −19.29128690203038494229598722261, −18.6714709607968281625472223308, −17.89036028979409505528407171824, −17.47125962748984940507055376842, −16.485648477042337130193258485830, −15.71088778861945494955010884178, −15.27930684986056911480190805274, −14.35316767060856104794113001233, −13.52849802105502353861152245105, −12.413569382596547386041035335649, −12.019602790875845400715700068123, −11.227854934496175133832511384481, −10.340471434805477112643125147772, −10.103633968423352003745270939530, −8.914320502007278248962607145345, −7.729703400160939955880816911681, −7.08478487618778537017831556754, −6.54089937846660600049810923096, −5.55466568872944452934389573283, −4.56935736102447389741280086622, −4.15050746349997543083185703961, −2.70367998020404802184028874316, −2.09397124943670735725132404083, −0.43327750467264365769784666267, 0.18370507618605457941402728296, 1.17067679336500280373333186579, 2.210766304474888157332708465523, 3.542557340213477794596296297393, 4.49902083119529203890915565630, 5.071445033661971025346204400415, 5.90460646546833701351004549369, 6.61889663265424576626874417163, 7.92216048912623648816688359605, 8.08182027894555249700385940340, 9.298996159124611734577703260835, 10.27501868115767007277439529452, 10.76708442192599270500897571677, 11.777947185904133683517456104554, 12.2868247394198352948915297141, 13.00543348932012600726782939859, 13.55913110901870624424329934847, 14.94840612356399737827950940634, 15.52768998900649959067447185677, 16.275940963392777153249567899549, 16.916376144050496881192426187128, 17.460318639131955334902651027504, 18.27502500036729932294765382159, 19.17004552086396020278745727863, 19.696601207347674314184724762050

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