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Properties

Label 1-1792-1792.11-r1-0-0
Degree $1$
Conductor $1792$
Sign $-0.0388 - 0.999i$
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  + (−0.582 − 0.812i)3-s + (−0.528 − 0.849i)5-s + (−0.321 + 0.946i)9-s + (0.935 + 0.352i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (−0.729 + 0.683i)19-s + (−0.896 + 0.442i)23-s + (−0.442 + 0.896i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.973 − 0.227i)37-s + ⋯
L(s)  = 1  + (−0.582 − 0.812i)3-s + (−0.528 − 0.849i)5-s + (−0.321 + 0.946i)9-s + (0.935 + 0.352i)11-s + (−0.881 + 0.471i)13-s + (−0.382 + 0.923i)15-s + (0.608 − 0.793i)17-s + (−0.729 + 0.683i)19-s + (−0.896 + 0.442i)23-s + (−0.442 + 0.896i)25-s + (0.956 − 0.290i)27-s + (−0.634 + 0.773i)29-s + (0.258 − 0.965i)31-s + (−0.258 − 0.965i)33-s + (−0.973 − 0.227i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.0388 - 0.999i$
Analytic conductor: \(192.577\)
Root analytic conductor: \(192.577\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (1:\ ),\ -0.0388 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5387260836 - 0.5600625754i\)
\(L(\frac12)\) \(\approx\) \(0.5387260836 - 0.5600625754i\)
\(L(1)\) \(\approx\) \(0.6590532040 - 0.2103207137i\)
\(L(1)\) \(\approx\) \(0.6590532040 - 0.2103207137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.582 - 0.812i)T \)
5 \( 1 + (-0.528 - 0.849i)T \)
11 \( 1 + (0.935 + 0.352i)T \)
13 \( 1 + (-0.881 + 0.471i)T \)
17 \( 1 + (0.608 - 0.793i)T \)
19 \( 1 + (-0.729 + 0.683i)T \)
23 \( 1 + (-0.896 + 0.442i)T \)
29 \( 1 + (-0.634 + 0.773i)T \)
31 \( 1 + (0.258 - 0.965i)T \)
37 \( 1 + (-0.973 - 0.227i)T \)
41 \( 1 + (-0.555 + 0.831i)T \)
43 \( 1 + (-0.995 - 0.0980i)T \)
47 \( 1 + (-0.130 + 0.991i)T \)
53 \( 1 + (-0.352 + 0.935i)T \)
59 \( 1 + (-0.0327 - 0.999i)T \)
61 \( 1 + (-0.910 - 0.412i)T \)
67 \( 1 + (0.812 - 0.582i)T \)
71 \( 1 + (0.980 + 0.195i)T \)
73 \( 1 + (-0.751 + 0.659i)T \)
79 \( 1 + (0.793 - 0.608i)T \)
83 \( 1 + (-0.290 + 0.956i)T \)
89 \( 1 + (-0.0654 - 0.997i)T \)
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

−20.06079067765993718942550337012, −19.54578228923498834319419219832, −18.80691245722391536790333495351, −17.83734372212260225584255907417, −17.1919809577027192670653992238, −16.61724127908985963526246563102, −15.686285767205740437732350867055, −14.99741099881117939417898501472, −14.62664922582036942162347655051, −13.711583275928272611785341270725, −12.37732576635287273381327619544, −11.93895881670350380287556654595, −11.18097402061700160094531229584, −10.33852882918708848473405965363, −10.01250633300621660690815340339, −8.86032907815696757413152348271, −8.13238712872910812074407010083, −6.98731823999667121133420194173, −6.42219842738396922857165180794, −5.561318921128258923017963494182, −4.59313646182499055552082802480, −3.772738248291301153894232004326, −3.21029870947513407679737344687, −2.000498637069844034041056821630, −0.48926650925951332165595814055, 0.29366952129854673892762195543, 1.421015076245400961696919808767, 2.00267711034483795715926922311, 3.427502026500549057627803228789, 4.4375452879072235629236240577, 5.08823110356912584735216593070, 5.9989338591798927601605241592, 6.860371196676208934527916976354, 7.61114376860774097977154764948, 8.22230502750426986970265687005, 9.26326953090529923969465251318, 9.91478466931399474204428827740, 11.140961277142856632902986062089, 11.84369100949059654715037855949, 12.259580451198663150872047970, 12.86923443107914013324645845785, 13.89247714423151945355374412388, 14.486484125969834760478026635884, 15.50933527227025051115755103919, 16.47479865152576002164976635775, 16.898934879118210458035041247639, 17.4206461541706484611756215712, 18.517057818597007821024468421762, 19.02695994678030824097115644296, 19.8529662659109367945256393447

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