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Properties

Label 1-1792-1792.787-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.348 - 0.937i$
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  + (0.0327 + 0.999i)3-s + (0.352 + 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (−0.773 + 0.634i)13-s + (−0.923 + 0.382i)15-s + (0.130 + 0.991i)17-s + (−0.812 − 0.582i)19-s + (0.659 − 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.412 + 0.910i)37-s + ⋯
L(s)  = 1  + (0.0327 + 0.999i)3-s + (0.352 + 0.935i)5-s + (−0.997 + 0.0654i)9-s + (−0.227 − 0.973i)11-s + (−0.773 + 0.634i)13-s + (−0.923 + 0.382i)15-s + (0.130 + 0.991i)17-s + (−0.812 − 0.582i)19-s + (0.659 − 0.751i)23-s + (−0.751 + 0.659i)25-s + (−0.0980 − 0.995i)27-s + (0.290 + 0.956i)29-s + (−0.965 − 0.258i)31-s + (0.965 − 0.258i)33-s + (−0.412 + 0.910i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.348 - 0.937i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.348 - 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1183792182 + 0.1703063032i\)
\(L(\frac12)\) \(\approx\) \(-0.1183792182 + 0.1703063032i\)
\(L(1)\) \(\approx\) \(0.7212340567 + 0.4002777616i\)
\(L(1)\) \(\approx\) \(0.7212340567 + 0.4002777616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.0327 + 0.999i)T \)
5 \( 1 + (0.352 + 0.935i)T \)
11 \( 1 + (-0.227 - 0.973i)T \)
13 \( 1 + (-0.773 + 0.634i)T \)
17 \( 1 + (0.130 + 0.991i)T \)
19 \( 1 + (-0.812 - 0.582i)T \)
23 \( 1 + (0.659 - 0.751i)T \)
29 \( 1 + (0.290 + 0.956i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (-0.412 + 0.910i)T \)
41 \( 1 + (0.195 - 0.980i)T \)
43 \( 1 + (-0.881 + 0.471i)T \)
47 \( 1 + (0.608 + 0.793i)T \)
53 \( 1 + (-0.973 + 0.227i)T \)
59 \( 1 + (0.162 - 0.986i)T \)
61 \( 1 + (-0.528 - 0.849i)T \)
67 \( 1 + (-0.999 + 0.0327i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (-0.896 - 0.442i)T \)
79 \( 1 + (-0.991 - 0.130i)T \)
83 \( 1 + (0.995 + 0.0980i)T \)
89 \( 1 + (0.321 - 0.946i)T \)
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

−19.786174818461671164051011751712, −18.98874326981182939362199047060, −18.08480274202256344970676565654, −17.53032169716881463467530242481, −16.98669916506547224380932848161, −16.159853621282482176154791730134, −15.11004799141538393751648222992, −14.48631991089375670055240224083, −13.43581799908166687319311672383, −13.06046941495542809172038685535, −12.22464984174016503845877624754, −11.884622171979892518404755093267, −10.652096571763466949731599900, −9.72692826471239777274106086691, −9.076841422763044868604559056916, −8.14441933120914506599328308185, −7.49226838213335161925470564455, −6.804590307745199814871161893629, −5.64551976744464259606999018479, −5.20039453240664319516060372695, −4.20140416867237096076613038079, −2.86317650435828133855093848279, −2.08322573465585717245286426952, −1.2726568236712394763310342539, −0.06885359250533733883191600788, 1.83065601170311704306164147440, 2.82819158464171523263889367429, 3.40735040771462474159180171839, 4.4071565650486202346088985249, 5.22439956695398686416030209625, 6.153682998105310040579499826600, 6.7549144255316376838673825776, 7.891113400154144541062268431570, 8.80892079632662065844160648785, 9.385558067080945472745734048966, 10.47445428977504882926509435169, 10.71422077694352038005814754752, 11.44119751062445282648209383617, 12.50050287365062255252099007375, 13.47659595075626038126804454194, 14.343056908327101629235897980826, 14.72279652218872088241171654414, 15.44191502398919080712260979011, 16.33340392418815726423497859582, 17.027722847119724102068499536616, 17.542742804106022294051474073711, 18.76301355828518151964974806700, 19.11128506119292018887817017540, 19.99261699332582142061909261612, 20.92319784332499379521566838781

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