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Properties

Label 1-1792-1792.173-r1-0-0
Degree $1$
Conductor $1792$
Sign $0.0857 + 0.996i$
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  + (0.528 + 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (0.634 + 0.773i)13-s + (0.923 − 0.382i)15-s + (0.793 − 0.608i)17-s + (0.412 + 0.910i)19-s + (0.321 + 0.946i)23-s + (−0.946 − 0.321i)25-s + (−0.995 + 0.0980i)27-s + (0.956 − 0.290i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.812 − 0.582i)37-s + ⋯
L(s)  = 1  + (0.528 + 0.849i)3-s + (0.162 − 0.986i)5-s + (−0.442 + 0.896i)9-s + (0.683 + 0.729i)11-s + (0.634 + 0.773i)13-s + (0.923 − 0.382i)15-s + (0.793 − 0.608i)17-s + (0.412 + 0.910i)19-s + (0.321 + 0.946i)23-s + (−0.946 − 0.321i)25-s + (−0.995 + 0.0980i)27-s + (0.956 − 0.290i)29-s + (−0.258 − 0.965i)31-s + (−0.258 + 0.965i)33-s + (0.812 − 0.582i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0857 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0857 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.0857 + 0.996i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ 0.0857 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.435435812 + 2.234853867i\)
\(L(\frac12)\) \(\approx\) \(2.435435812 + 2.234853867i\)
\(L(1)\) \(\approx\) \(1.426193373 + 0.4499311434i\)
\(L(1)\) \(\approx\) \(1.426193373 + 0.4499311434i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.528 + 0.849i)T \)
5 \( 1 + (0.162 - 0.986i)T \)
11 \( 1 + (0.683 + 0.729i)T \)
13 \( 1 + (0.634 + 0.773i)T \)
17 \( 1 + (0.793 - 0.608i)T \)
19 \( 1 + (0.412 + 0.910i)T \)
23 \( 1 + (0.321 + 0.946i)T \)
29 \( 1 + (0.956 - 0.290i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.812 - 0.582i)T \)
41 \( 1 + (-0.195 + 0.980i)T \)
43 \( 1 + (0.471 + 0.881i)T \)
47 \( 1 + (0.991 - 0.130i)T \)
53 \( 1 + (-0.729 + 0.683i)T \)
59 \( 1 + (0.352 + 0.935i)T \)
61 \( 1 + (-0.0327 - 0.999i)T \)
67 \( 1 + (-0.849 + 0.528i)T \)
71 \( 1 + (0.555 - 0.831i)T \)
73 \( 1 + (-0.0654 - 0.997i)T \)
79 \( 1 + (-0.608 + 0.793i)T \)
83 \( 1 + (-0.0980 + 0.995i)T \)
89 \( 1 + (-0.659 - 0.751i)T \)
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

−19.72509024421245623437906975179, −18.96719469772449242310830440815, −18.5840575611764040126072173975, −17.71406192157001575529793372207, −17.22673471507612618358422961017, −16.04854909621765780311558511021, −15.24438773437392154025622608978, −14.42774251981950175321829173049, −14.03852811383720809795481085240, −13.25707382043280367746331214649, −12.457004043971484529049768458551, −11.65138262341972612542855659545, −10.83515648581698613095017430900, −10.163361667731973513332198412986, −9.008996510168102186662912936288, −8.4545859962689705865944457023, −7.58319713422668870318030840339, −6.77764970356099422647601862812, −6.23313693678959430370110848652, −5.408627131809694682334366652208, −3.883556389092529643485742416894, −3.16243718040595700036618712256, −2.59392285336568276184552986279, −1.3794812061955291226328928590, −0.609396497169162318923660627080, 1.05033855358423043623407372471, 1.82567582722303715598536043213, 2.99050710180931620890457741159, 4.02903279976806928137591336559, 4.44960315233664487555051486900, 5.41347618573891056169485141353, 6.151306805595423927658910964, 7.49733653999293957411741920826, 8.09392519979073132417578487107, 9.14579562966212449143840496534, 9.45195272192146880815412243837, 10.107447324827301910638730547656, 11.30847556307435483134708881009, 11.891717618704561170808191086084, 12.779970090899011112012979975817, 13.74084942040302695993406931947, 14.18334672079615104148835314601, 15.06576060283777656723996019465, 15.861777213724775132061628894204, 16.523272101421552850733697342867, 16.937046604575014456554818090701, 17.91764812562688349378473194857, 18.877386492808699668031789487537, 19.7181483383212651159953760263, 20.24352091088336948175577360951

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