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Properties

Label 1-792-792.787-r0-0-0
Degree $1$
Conductor $792$
Sign $0.999 + 0.0129i$
Analytic cond. $3.67803$
Root an. cond. $3.67803$
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  + (0.104 − 0.994i)5-s + (0.669 + 0.743i)7-s + (0.913 + 0.406i)13-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)29-s + (−0.913 − 0.406i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + (−0.669 + 0.743i)41-s + (0.5 − 0.866i)43-s + (0.978 + 0.207i)47-s + (−0.104 + 0.994i)49-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)5-s + (0.669 + 0.743i)7-s + (0.913 + 0.406i)13-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)19-s + (0.5 + 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.669 + 0.743i)29-s + (−0.913 − 0.406i)31-s + (0.809 − 0.587i)35-s + (−0.309 + 0.951i)37-s + (−0.669 + 0.743i)41-s + (0.5 − 0.866i)43-s + (0.978 + 0.207i)47-s + (−0.104 + 0.994i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(792\)    =    \(2^{3} \cdot 3^{2} \cdot 11\)
Sign: $0.999 + 0.0129i$
Analytic conductor: \(3.67803\)
Root analytic conductor: \(3.67803\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{792} (787, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 792,\ (0:\ ),\ 0.999 + 0.0129i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.707169952 + 0.01102922681i\)
\(L(\frac12)\) \(\approx\) \(1.707169952 + 0.01102922681i\)
\(L(1)\) \(\approx\) \(1.245091517 - 0.04278525111i\)
\(L(1)\) \(\approx\) \(1.245091517 - 0.04278525111i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.978 + 0.207i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.978 + 0.207i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.104 - 0.994i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−22.511777810078961151271604649077, −21.2839875615270004522837421956, −20.86312833831706468760324310019, −19.95574252763916970239557843303, −18.87957955545523403567395411197, −18.34933086939182361106448782508, −17.56815047129539718153325820525, −16.70063893806348463888989509294, −15.8135351447998272525851762861, −14.78637846304490878862385349796, −14.222341209470980600178576914601, −13.55592861574640297776451453714, −12.42151962250397248850341914926, −11.45120721663308953976004628511, −10.58368077080147702785633098950, −10.258780127287494054035526134356, −8.92219258778821454811321985870, −7.89839758328554461288903830063, −7.25464850021977284800830891858, −6.26760737318280454296868073019, −5.38067802630782281870628264485, −4.11636764023379522134874481335, −3.36119081028908944606784072330, −2.22165229912236240795952806029, −0.99879638313333072532501573148, 1.15035107022331879078628972034, 1.94959553890068217611804014160, 3.32756629487280223622573048233, 4.44894972484048711556504062654, 5.28588193400434375205533590774, 5.99002678336917068396236929254, 7.235545595862921447037767695339, 8.40624882426964990389060854909, 8.76478211042123402712578332756, 9.68544654320172939429082647593, 10.89552090135146639937292179650, 11.663902428326825047275594416202, 12.457047394660327175500541357409, 13.26234262640153371902912792873, 14.08344155776164334566392223112, 15.14738509841978039495232059834, 15.75716121267961814551180146055, 16.737703711852311657444601848588, 17.37817468496167433186341180162, 18.28123940830824901217284485447, 19.048265123662403205186970532889, 19.99479155198564789963369688211, 20.80286097213617498151734383543, 21.44545073162234031773300096260, 21.99239157279903923291751359844

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