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Properties

Label 1-1792-1792.131-r0-0-0
Degree $1$
Conductor $1792$
Sign $0.780 - 0.624i$
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  + (0.986 − 0.162i)3-s + (0.227 + 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (0.0980 − 0.995i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.528 − 0.849i)37-s + ⋯
L(s)  = 1  + (0.986 − 0.162i)3-s + (0.227 + 0.973i)5-s + (0.946 − 0.321i)9-s + (0.412 − 0.910i)11-s + (−0.290 − 0.956i)13-s + (0.382 + 0.923i)15-s + (0.608 + 0.793i)17-s + (−0.0327 − 0.999i)19-s + (0.442 − 0.896i)23-s + (−0.896 + 0.442i)25-s + (0.881 − 0.471i)27-s + (0.0980 − 0.995i)29-s + (−0.258 − 0.965i)31-s + (0.258 − 0.965i)33-s + (0.528 − 0.849i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $0.780 - 0.624i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ 0.780 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.386116977 - 0.8372082093i\)
\(L(\frac12)\) \(\approx\) \(2.386116977 - 0.8372082093i\)
\(L(1)\) \(\approx\) \(1.593652939 - 0.1579350812i\)
\(L(1)\) \(\approx\) \(1.593652939 - 0.1579350812i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.986 - 0.162i)T \)
5 \( 1 + (0.227 + 0.973i)T \)
11 \( 1 + (0.412 - 0.910i)T \)
13 \( 1 + (-0.290 - 0.956i)T \)
17 \( 1 + (0.608 + 0.793i)T \)
19 \( 1 + (-0.0327 - 0.999i)T \)
23 \( 1 + (0.442 - 0.896i)T \)
29 \( 1 + (0.0980 - 0.995i)T \)
31 \( 1 + (-0.258 - 0.965i)T \)
37 \( 1 + (0.528 - 0.849i)T \)
41 \( 1 + (-0.831 + 0.555i)T \)
43 \( 1 + (-0.634 + 0.773i)T \)
47 \( 1 + (-0.130 - 0.991i)T \)
53 \( 1 + (-0.910 - 0.412i)T \)
59 \( 1 + (0.683 + 0.729i)T \)
61 \( 1 + (0.935 + 0.352i)T \)
67 \( 1 + (0.162 + 0.986i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (-0.659 + 0.751i)T \)
79 \( 1 + (-0.793 - 0.608i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (-0.997 - 0.0654i)T \)
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

−20.48261203713590507542828343715, −19.64898703789917367788529033151, −18.980709527520146424891276792238, −18.18213994694247351553973024395, −17.20655841382538426144920477995, −16.53657580482478758507010180017, −15.89571100776260797682194031554, −15.04950484279726969889131412183, −14.1800452848824908455454330953, −13.86274104105864489728235857132, −12.74024418247150131537525339454, −12.33147273922058395912973319446, −11.44207288639373180165488201661, −10.071345704661907880414232495708, −9.65273230444822150624044730283, −8.99524480836750165945670344395, −8.270544819252224770608411871059, −7.3728133041600267105245280775, −6.72681010232180430246559710691, −5.33542748793322819101298783834, −4.73615427269665202587352091385, −3.909890449830109676524723123440, −3.03647821336307306663644436962, −1.76651625042924809115336324040, −1.43177739129836458011252921779, 0.795526737416291874433977067934, 2.07234354637108292861074115778, 2.84953989288654529979419100118, 3.42824938213910396364017961705, 4.33773506610609984300168603065, 5.6422479996371050132726110958, 6.41236018505001235996936391456, 7.18863838052512054722637422470, 8.022962367768012892998745185378, 8.61105806456478342032880298808, 9.639205727593260789987150692, 10.21756552330229116164998263694, 11.040166756277902430569285419182, 11.86076918862147213431201427968, 13.15263502294594690446787248501, 13.260899355596528177737211949921, 14.442794872634558888532403094991, 14.767600962848590394675044970012, 15.396005774517348814449266938297, 16.39237960739310201116462908130, 17.32888935575196317542275449746, 18.061397020370049999871453618280, 18.86152458796220099472004487673, 19.29064319983229637276962652781, 19.9960189144223245609934546226

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