Properties

Label 1-1792-1792.555-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.996 + 0.0878i$
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  + (0.352 + 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.849 + 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯
L(s)  = 1  + (0.352 + 0.935i)3-s + (−0.729 − 0.683i)5-s + (−0.751 + 0.659i)9-s + (0.582 − 0.812i)11-s + (0.290 − 0.956i)13-s + (0.382 − 0.923i)15-s + (0.991 + 0.130i)17-s + (0.849 + 0.528i)19-s + (−0.997 − 0.0654i)23-s + (0.0654 + 0.997i)25-s + (−0.881 − 0.471i)27-s + (0.0980 + 0.995i)29-s + (−0.965 + 0.258i)31-s + (0.965 + 0.258i)33-s + (−0.999 + 0.0327i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.996 + 0.0878i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (555, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.996 + 0.0878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.057964394 + 0.09052529530i\)
\(L(\frac12)\) \(\approx\) \(2.057964394 + 0.09052529530i\)
\(L(1)\) \(\approx\) \(1.092131483 + 0.1390622640i\)
\(L(1)\) \(\approx\) \(1.092131483 + 0.1390622640i\)

Euler product

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

−19.73245110526303508959587037129, −19.38485115602494575873847216454, −18.47921713486891062539739485656, −18.11195908478737915724068709485, −17.20393130320950193780233527817, −16.30935672926218617721695300831, −15.46901970587906034765787885972, −14.61901944557344160166401362916, −14.1718239584719619523751328978, −13.44353258860423669609395465798, −12.35146574181763523468106881308, −11.82735004755851858228682577517, −11.37822183479792108138239983899, −10.14582682239165031491835754614, −9.36536366223010671527833821367, −8.495518168064076505742897626615, −7.57290757575124965202383396099, −7.16635425160900559314210316119, −6.43806218577969029942521905266, −5.49899121432348358187869427412, −4.13973504675844376778315372309, −3.57878095840908548924855364429, −2.52359913458481631152519233538, −1.73467675582837605160375615093, −0.652982314084997204725544984462, 0.54516882413524850198364140118, 1.55065081880333570792482378755, 3.1550005561978488146969600502, 3.5204881182617146509302673210, 4.31504050842479673379557819394, 5.45902723022105962407204580808, 5.70989798643250088453769529296, 7.29251824157117508248816148126, 8.08787337100910291294793648889, 8.62308374174762217761780116950, 9.37073503501188413329343103725, 10.26553460535844981501162713255, 10.92520044915077635486170909714, 11.83874364473455270807052179644, 12.40732343306518337706901102467, 13.47459237564248126284401456590, 14.25409494055924838098025757761, 14.858057460299395634534300585828, 15.82156241152580755258438395086, 16.23756014532591995933279923290, 16.75588968267563670494598538856, 17.7309444912412289276885372372, 18.754431349030090908917546823430, 19.47345686187989798520063738535, 20.28635739042102781234292062497

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