Properties

Label 1-1792-1792.1299-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.877 + 0.480i$
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  + (−0.0327 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.877 + 0.480i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.877 + 0.480i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5724035517 + 0.1465030837i\)
\(L(\frac12)\) \(\approx\) \(0.5724035517 + 0.1465030837i\)
\(L(1)\) \(\approx\) \(0.7632449782 - 0.4485703727i\)
\(L(1)\) \(\approx\) \(0.7632449782 - 0.4485703727i\)

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

−19.95368334458317477419031170237, −19.30722760249270774890236887470, −18.55132768666332534666230409529, −17.49166721036842178135539052246, −17.25926693655488952321594156484, −15.81875808014020323339948823232, −15.687330031568282812393755693751, −14.942844313415716279464711833366, −14.176704523552078396007663519397, −13.48854679263107805954747655622, −12.333314986615570603938131200160, −11.39988518494639713903108163707, −11.05339199505095015633966492280, −10.10561590103722332666402222110, −9.61319756482054173215076207144, −8.6454757523000087477740168431, −7.803790085177221083211051705, −6.874706370892506602496049931370, −6.137459297057878008760797854482, −5.1403712302354983973184876973, −4.25523288975130253808389715172, −3.60848385861101994675919901298, −2.776797508479410164473602032962, −1.74558119027978760732562083756, −0.123672266365836927758310486754, 0.94956860436844443479072516957, 1.304425755448300308638789260609, 2.85194171245749852703842893251, 3.35538755416175875879650630266, 4.78943531951177418460284605651, 5.405133106802460282572999186233, 6.269312679176383960131788470486, 7.10225816519411693144674932494, 8.19852152433231493261960259325, 8.33822319022580841982141718382, 9.25393742192187459331042721004, 10.408660412646705425207486592981, 11.38392687832497769213594645662, 11.868110721212104111697831480197, 12.683985700786262372301251401074, 13.42066270903088857599970078411, 13.78909969017309097146814834054, 14.84214046884823540803117071439, 15.90157712217792111403188026569, 16.40633516210602814605777955737, 17.08325523492897679267383211305, 18.17742594088816498346691310313, 18.432103481048536632972713961661, 19.35847830232020929927693830135, 20.07978421817029963812636645499

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