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Properties

Label 1-1792-1792.227-r0-0-0
Degree $1$
Conductor $1792$
Sign $-0.810 - 0.585i$
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  + (0.910 − 0.412i)3-s + (0.0327 − 0.999i)5-s + (0.659 − 0.751i)9-s + (−0.986 − 0.162i)11-s + (0.471 − 0.881i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.973 − 0.227i)19-s + (−0.0654 − 0.997i)23-s + (−0.997 − 0.0654i)25-s + (0.290 − 0.956i)27-s + (−0.773 + 0.634i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.729 + 0.683i)37-s + ⋯
L(s)  = 1  + (0.910 − 0.412i)3-s + (0.0327 − 0.999i)5-s + (0.659 − 0.751i)9-s + (−0.986 − 0.162i)11-s + (0.471 − 0.881i)13-s + (−0.382 − 0.923i)15-s + (0.991 − 0.130i)17-s + (−0.973 − 0.227i)19-s + (−0.0654 − 0.997i)23-s + (−0.997 − 0.0654i)25-s + (0.290 − 0.956i)27-s + (−0.773 + 0.634i)29-s + (0.965 + 0.258i)31-s + (−0.965 + 0.258i)33-s + (−0.729 + 0.683i)37-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1792\)    =    \(2^{8} \cdot 7\)
Sign: $-0.810 - 0.585i$
Analytic conductor: \(8.32201\)
Root analytic conductor: \(8.32201\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1792} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1792,\ (0:\ ),\ -0.810 - 0.585i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5601443174 - 1.731346208i\)
\(L(\frac12)\) \(\approx\) \(0.5601443174 - 1.731346208i\)
\(L(1)\) \(\approx\) \(1.149481242 - 0.6623483780i\)
\(L(1)\) \(\approx\) \(1.149481242 - 0.6623483780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.910 - 0.412i)T \)
5 \( 1 + (0.0327 - 0.999i)T \)
11 \( 1 + (-0.986 - 0.162i)T \)
13 \( 1 + (0.471 - 0.881i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (-0.973 - 0.227i)T \)
23 \( 1 + (-0.0654 - 0.997i)T \)
29 \( 1 + (-0.773 + 0.634i)T \)
31 \( 1 + (0.965 + 0.258i)T \)
37 \( 1 + (-0.729 + 0.683i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (-0.0980 - 0.995i)T \)
47 \( 1 + (0.793 - 0.608i)T \)
53 \( 1 + (-0.162 + 0.986i)T \)
59 \( 1 + (-0.528 + 0.849i)T \)
61 \( 1 + (-0.582 + 0.812i)T \)
67 \( 1 + (0.412 + 0.910i)T \)
71 \( 1 + (-0.980 + 0.195i)T \)
73 \( 1 + (0.946 - 0.321i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (0.956 - 0.290i)T \)
89 \( 1 + (0.896 - 0.442i)T \)
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

−20.693916029529453434818432784482, −19.615817395953212633669854898429, −18.91534345086881102518241262534, −18.68419426017514965694202426268, −17.65102512674219472449500245817, −16.712928032832611022644097602223, −15.84189329306080931376873604998, −15.26040181073516621869374774398, −14.625535845181327823654492209410, −13.88333281758531143069347382617, −13.35546691463520570403311607370, −12.38147410002013855283801150792, −11.26756826837044734659296260935, −10.68453554420263016691453260146, −9.8628025047528343420535236223, −9.374937126041939408821223606005, −8.10448291076075383440719488840, −7.81671785480334815478638621046, −6.81322264145132001275239558753, −5.95520449942530426086187826216, −4.89967814229059495498447448673, −3.89782343662415004491866717571, −3.29901988639199891325205636697, −2.37854207460622078939487270860, −1.679991211491317740303314981663, 0.536317611336919271641651913172, 1.51312668067554735697915661186, 2.51022382901778179463278785242, 3.32352030064007317870880330916, 4.26669849388909706545595745203, 5.21477585286459325831796887213, 5.97182870352064509107618130831, 7.10511082995360486793421868146, 7.92943645410129464377574988113, 8.51049127388849818983027855500, 9.00121305239780708508185268649, 10.179250833099196617450626941493, 10.60246242367331864903598153677, 12.192954815799651876788214180338, 12.39810430652069507631587986352, 13.42817187446475390728801327294, 13.61361174951355825561025449493, 14.82380169646569120051866618402, 15.410076116997830767095796902397, 16.13414399961601264793841476066, 16.97902607177152567067251936033, 17.7699780391762292052422422153, 18.68995799731673384394153403978, 19.04090728381890056137212861212, 20.15512699193488358429435465937

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