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Properties

Label 2-3311-3311.475-c0-0-0
Degree $2$
Conductor $3311$
Sign $0.875 + 0.483i$
Analytic cond. $1.65240$
Root an. cond. $1.28545$
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  + (−0.202 − 1.49i)2-s + (−1.22 + 0.337i)4-s + (0.809 + 0.587i)7-s + (0.158 + 0.371i)8-s + (−0.936 + 0.351i)9-s + (−0.134 + 0.990i)11-s + (0.713 − 1.32i)14-s + (−0.566 + 0.338i)16-s + (0.713 + 1.32i)18-s + 1.50·22-s + (0.490 + 0.614i)23-s + (−0.983 + 0.178i)25-s + (−1.18 − 0.445i)28-s + (1.84 + 0.334i)29-s + (0.871 + 1.09i)32-s + ⋯
L(s)  = 1  + (−0.202 − 1.49i)2-s + (−1.22 + 0.337i)4-s + (0.809 + 0.587i)7-s + (0.158 + 0.371i)8-s + (−0.936 + 0.351i)9-s + (−0.134 + 0.990i)11-s + (0.713 − 1.32i)14-s + (−0.566 + 0.338i)16-s + (0.713 + 1.32i)18-s + 1.50·22-s + (0.490 + 0.614i)23-s + (−0.983 + 0.178i)25-s + (−1.18 − 0.445i)28-s + (1.84 + 0.334i)29-s + (0.871 + 1.09i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3311\)    =    \(7 \cdot 11 \cdot 43\)
Sign: $0.875 + 0.483i$
Analytic conductor: \(1.65240\)
Root analytic conductor: \(1.28545\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3311} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3311,\ (\ :0),\ 0.875 + 0.483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9515933273\)
\(L(\frac12)\) \(\approx\) \(0.9515933273\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
43 \( 1 + (0.550 + 0.834i)T \)
good2 \( 1 + (0.202 + 1.49i)T + (-0.963 + 0.266i)T^{2} \)
3 \( 1 + (0.936 - 0.351i)T^{2} \)
5 \( 1 + (0.983 - 0.178i)T^{2} \)
13 \( 1 + (0.473 + 0.880i)T^{2} \)
17 \( 1 + (0.473 - 0.880i)T^{2} \)
19 \( 1 + (0.995 + 0.0896i)T^{2} \)
23 \( 1 + (-0.490 - 0.614i)T + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (-1.84 - 0.334i)T + (0.936 + 0.351i)T^{2} \)
31 \( 1 + (0.963 - 0.266i)T^{2} \)
37 \( 1 + (1.08 - 1.48i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.550 - 0.834i)T^{2} \)
47 \( 1 + (0.995 + 0.0896i)T^{2} \)
53 \( 1 + (-1.98 - 0.178i)T + (0.983 + 0.178i)T^{2} \)
59 \( 1 + (0.691 + 0.722i)T^{2} \)
61 \( 1 + (0.963 + 0.266i)T^{2} \)
67 \( 1 + (-0.167 + 0.209i)T + (-0.222 - 0.974i)T^{2} \)
71 \( 1 + (0.668 - 1.78i)T + (-0.753 - 0.657i)T^{2} \)
73 \( 1 + (-0.134 + 0.990i)T^{2} \)
79 \( 1 + (-1.58 - 0.515i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.963 - 0.266i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.753 + 0.657i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.694260264679079588572372878679, −8.490711622865207359949909135615, −7.44977985633490374413050696408, −6.51132937692505747228839765909, −5.35469527274434728575159484514, −4.85353438818136418390278317404, −3.85897737210185054629166313174, −2.85144791240976464353653141574, −2.23052073484110709263247182507, −1.35066563039276966294507278742, 0.64156341193645344857552346874, 2.39133399603589592381918092260, 3.52937009179138964532822862439, 4.57505826942918313858079900259, 5.31228429122094239784431547754, 5.99759902317792349125957602125, 6.60727315165560491054444899747, 7.42594892710212626215963809119, 8.151830613891668271185973209478, 8.552192525442552382329065861577

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