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Properties

Label 1-42e2-1764.1483-r0-0-0
Degree $1$
Conductor $1764$
Sign $0.720 + 0.693i$
Analytic cond. $8.19198$
Root an. cond. $8.19198$
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  + (0.733 + 0.680i)5-s + (−0.826 − 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.623 − 0.781i)17-s + 19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (0.826 + 0.563i)47-s + (0.623 − 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)5-s + (−0.826 − 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.623 − 0.781i)17-s + 19-s + (−0.365 − 0.930i)23-s + (0.0747 + 0.997i)25-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (0.733 + 0.680i)41-s + (0.733 − 0.680i)43-s + (0.826 + 0.563i)47-s + (0.623 − 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.720 + 0.693i$
Analytic conductor: \(8.19198\)
Root analytic conductor: \(8.19198\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1764} (1483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1764,\ (0:\ ),\ 0.720 + 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581827045 + 0.6371342491i\)
\(L(\frac12)\) \(\approx\) \(1.581827045 + 0.6371342491i\)
\(L(1)\) \(\approx\) \(1.164039655 + 0.1792538152i\)
\(L(1)\) \(\approx\) \(1.164039655 + 0.1792538152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (-0.0747 + 0.997i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (0.733 + 0.680i)T \)
43 \( 1 + (0.733 - 0.680i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (0.623 - 0.781i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.0747 + 0.997i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−20.215866770630441766582329326579, −19.66412690859471835132642930077, −18.47330728453421087776164729197, −17.724114349555521696226524657812, −17.514476110983890479395204995118, −16.398696701279520001818380399999, −15.77321802974690147506753348232, −15.05165796505496363333317517074, −14.12167456648492790129135082909, −13.280980104751730633717188213343, −12.82006953030236280494709677437, −12.135215575279394759476434340424, −10.97927149475587123294718401, −10.310865074848163048822077609274, −9.55974719546741145624004220363, −8.87186644060426231907965398274, −7.877593515261760981754680961044, −7.32730021343939976549454531333, −6.00283983426022640952405185421, −5.52181888753632717762013983598, −4.74584554474872836712505791371, −3.73377866293872889343058676375, −2.60510516412863464535111766956, −1.84341633266302484384517228271, −0.72021895421966020931149429982, 0.9799008877740635377313063045, 2.34752721126705154274164768402, 2.69859254961619208344797450801, 3.886605949290217499472641267778, 4.90496794308856519107830467179, 5.70512520378235100847508831185, 6.5410844071509899425440583495, 7.19024187126080559475922590481, 8.13405729261486602990341561344, 9.11450991319400122023862325488, 9.744653594154240590605650006715, 10.57169391780470163047190735224, 11.25003482850491395342029308835, 11.99462320987615449935306017502, 13.06805032426927097926762449665, 13.82257807311227280905302990045, 14.14394537514923878699906884650, 15.121743101338868644946779757522, 16.02431035984206721444020682803, 16.518013087975320830197982317613, 17.56586508499520704183762454005, 18.21890041086467902073048510841, 18.66646238446302955922108470231, 19.49456207026979793560371755371, 20.48192690075085303123729720334

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