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Properties

Label 2-3267-27.11-c0-0-0
Degree $2$
Conductor $3267$
Sign $-0.396 + 0.918i$
Analytic cond. $1.63044$
Root an. cond. $1.27688$
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  + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.673 + 1.85i)5-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)12-s + (−0.673 − 1.85i)15-s + (−0.939 + 0.342i)16-s + (−1.93 − 0.342i)20-s + (−1.70 + 0.300i)23-s + (−2.20 − 1.85i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 1.50i)31-s + 36-s + (0.173 + 0.300i)37-s + (1.70 + 0.984i)45-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.673 + 1.85i)5-s + (0.173 − 0.984i)9-s + (−0.766 − 0.642i)12-s + (−0.673 − 1.85i)15-s + (−0.939 + 0.342i)16-s + (−1.93 − 0.342i)20-s + (−1.70 + 0.300i)23-s + (−2.20 − 1.85i)25-s + (0.500 + 0.866i)27-s + (−0.266 − 1.50i)31-s + 36-s + (0.173 + 0.300i)37-s + (1.70 + 0.984i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.396 + 0.918i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (848, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ -0.396 + 0.918i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4141741119\)
\(L(\frac12)\) \(\approx\) \(0.4141741119\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T^{2} \)
5 \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \)
7 \( 1 + (-0.939 - 0.342i)T^{2} \)
13 \( 1 + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 - 0.984i)T^{2} \)
31 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
37 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
53 \( 1 - 1.28iT - T^{2} \)
59 \( 1 + (0.439 - 1.20i)T + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.939 - 0.342i)T^{2} \)
67 \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.495648896886032071416134327814, −8.463513630631696319323001484642, −7.43730051076813037369158896490, −7.37735689023851735278019200913, −6.22354924734362051816494541778, −5.92802653320564818552787770780, −4.27736987807645307424078805091, −3.98175038357704093492069605889, −3.12769073541937506830247325078, −2.34089851591561749872558152667, 0.28202531224940203087612249285, 1.31079925555514298866661564220, 2.06246410010373805515962438385, 3.90952031118968737506283365658, 4.73545157711655159260924214821, 5.27905137393718108484681144092, 5.88223317118208659174833851725, 6.72222024629403536965585664842, 7.62715773457759367506961474012, 8.285232498524436230619196719861

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