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Properties

Label 1-1792-1792.1483-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.0245 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0980 − 0.995i)3-s + (−0.881 − 0.471i)5-s + (−0.980 + 0.195i)9-s + (0.634 + 0.773i)11-s + (0.471 + 0.881i)13-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.290 − 0.956i)19-s + (−0.831 − 0.555i)23-s + (0.555 + 0.831i)25-s + (0.290 + 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + (0.956 − 0.290i)37-s + ⋯
L(s)  = 1  + (−0.0980 − 0.995i)3-s + (−0.881 − 0.471i)5-s + (−0.980 + 0.195i)9-s + (0.634 + 0.773i)11-s + (0.471 + 0.881i)13-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)17-s + (0.290 − 0.956i)19-s + (−0.831 − 0.555i)23-s + (0.555 + 0.831i)25-s + (0.290 + 0.956i)27-s + (−0.773 − 0.634i)29-s + (−0.707 − 0.707i)31-s + (0.707 − 0.707i)33-s + (0.956 − 0.290i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0245 + 0.999i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.0245 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1432917209 + 0.1468525050i\)
\(L(\frac12)\) \(\approx\) \(0.1432917209 + 0.1468525050i\)
\(L(1)\) \(\approx\) \(0.6806930422 - 0.2318435394i\)
\(L(1)\) \(\approx\) \(0.6806930422 - 0.2318435394i\)

Euler product

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

−20.12082551594348069769326490069, −19.414625553746318759758317344744, −18.6054998427977770143943792959, −17.76896817352649244241848831677, −16.91569027262115624303056306748, −16.14060211251852949143742534723, −15.72856189496267541430629548073, −14.78201137502982287312992993449, −14.471423498638407712572552637083, −13.41840296199569139212507410491, −12.35577826293383880284028261478, −11.608769012570284559827327016639, −10.92957763174666856638344661794, −10.419521770186639119159048073693, −9.52264200927108437756434367159, −8.448659131673549481984273090754, −8.17277239926177031206502914677, −6.97097881104953244211197955152, −5.98073527606134460451151039398, −5.41761634905001848836658628109, −4.15769349167778289530876605026, −3.60759012607366502969755591516, −3.13444987828021476941636457568, −1.61226623318851123299147101382, −0.08057125193827272518997482563, 1.11838595783145517234466167675, 2.00171514104915582706157326907, 3.008333209264268707094689869216, 4.20897983770954700090579713490, 4.7270727762227188999754611580, 5.96026461201874026668427226275, 6.744505244560517012028689495347, 7.41921587813887879040482137294, 8.06180496190337598591715318467, 9.055263303190326879634162231445, 9.50712812219985886888894371355, 11.09043348054412650552989364149, 11.56038602641489713572835781262, 12.05784953863838872193884064158, 12.994378328750026380229973837155, 13.49430087717953289893343844236, 14.48345033262286492258465618143, 15.10940091059589700311499115853, 16.23211140513962619506869071231, 16.585021786325121286953512969422, 17.60359917652637323197430436657, 18.22875625612204195293653463000, 18.920812657692667557661423174665, 19.73625891869566498314111594724, 20.14716022647885613400764083155

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