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Properties

Label 1-1792-1792.451-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.585 + 0.810i$
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  + (0.935 + 0.352i)3-s + (0.683 + 0.729i)5-s + (0.751 + 0.659i)9-s + (−0.812 + 0.582i)11-s + (−0.956 + 0.290i)13-s + (0.382 + 0.923i)15-s + (−0.991 + 0.130i)17-s + (0.528 + 0.849i)19-s + (0.997 − 0.0654i)23-s + (−0.0654 + 0.997i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.0327 − 0.999i)37-s + ⋯
L(s)  = 1  + (0.935 + 0.352i)3-s + (0.683 + 0.729i)5-s + (0.751 + 0.659i)9-s + (−0.812 + 0.582i)11-s + (−0.956 + 0.290i)13-s + (0.382 + 0.923i)15-s + (−0.991 + 0.130i)17-s + (0.528 + 0.849i)19-s + (0.997 − 0.0654i)23-s + (−0.0654 + 0.997i)25-s + (0.471 + 0.881i)27-s + (−0.995 − 0.0980i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (0.0327 − 0.999i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.585 + 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.585 + 0.810i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.585 + 0.810i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9285632291 + 1.816760934i\)
\(L(\frac12)\) \(\approx\) \(0.9285632291 + 1.816760934i\)
\(L(1)\) \(\approx\) \(1.290142243 + 0.6269137156i\)
\(L(1)\) \(\approx\) \(1.290142243 + 0.6269137156i\)

Euler product

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

−19.84855882210163365850593610117, −19.48097220645401065855164759888, −18.36681125106829430081569255005, −17.9208469084222868464254041034, −17.03901032604379097629980888887, −16.27505905481693999166292927167, −15.30525960804929639963191919410, −14.88211468431548428562308479796, −13.6163227734094037944273571231, −13.440414948689992285832944737495, −12.80430648073901454991767162388, −11.8762474274246834486015430962, −10.876972636854743944462261169667, −9.8472658734155218535605603052, −9.336629583976819797028089109414, −8.58107836870413030775876167770, −7.88181791488341320840851605734, −7.03871567535087034987565947725, −6.16949068943908619506783779633, −5.05178901125212175337583107833, −4.57738143269227510294215941458, −3.13454418575005584488724995626, −2.603460003314967859525410189418, −1.679384683801844991292361599922, −0.57221988025856666145342604206, 1.659411332273217632442415267397, 2.39891931479682808267434769890, 2.97165084450550163052829945950, 4.05600520447153781752715901606, 4.91315326577510927028373313787, 5.75246292832864650452889542121, 7.031669394008217697987012855252, 7.344684864135989956966008243152, 8.36568202243455947504803122796, 9.321766412276576267237841271595, 9.82894387601829914480306009384, 10.51369604168494211879842676313, 11.25031725796396791392820757957, 12.528371885113295972381027624411, 13.13228377334624398222006372016, 13.947948120007361249211935659432, 14.52489123431357926763947715276, 15.199430297644458637502698220586, 15.77527791150569531017097842629, 16.84142646586555472263349420874, 17.572887517250826597630005602808, 18.40860313023283607458224472793, 18.971525723740063703902095472267, 19.7555110781817242625674328781, 20.54376922344124198192312025398

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