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Properties

Label 2-3267-27.23-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.993 + 0.116i$
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  + (0.939 − 0.342i)3-s + (0.766 + 0.642i)4-s + (−1.26 − 0.223i)5-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)12-s + (−1.26 + 0.223i)15-s + (0.173 + 0.984i)16-s + (−0.826 − 0.984i)20-s + (1.11 − 1.32i)23-s + (0.613 + 0.223i)25-s + (0.500 − 0.866i)27-s + (1.43 + 1.20i)31-s + 36-s + (0.766 − 1.32i)37-s + (−1.11 + 0.642i)45-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)3-s + (0.766 + 0.642i)4-s + (−1.26 − 0.223i)5-s + (0.766 − 0.642i)9-s + (0.939 + 0.342i)12-s + (−1.26 + 0.223i)15-s + (0.173 + 0.984i)16-s + (−0.826 − 0.984i)20-s + (1.11 − 1.32i)23-s + (0.613 + 0.223i)25-s + (0.500 − 0.866i)27-s + (1.43 + 1.20i)31-s + 36-s + (0.766 − 1.32i)37-s + (−1.11 + 0.642i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.993 + 0.116i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (1211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.993 + 0.116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.774105816\)
\(L(\frac12)\) \(\approx\) \(1.774105816\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 \)
good2 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (1.26 + 0.223i)T + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.766 - 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-1.11 + 1.32i)T + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
53 \( 1 + 0.684iT - T^{2} \)
59 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (1.70 + 0.984i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.766 + 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.603832042936494714814497926793, −8.036161870456436092535755129015, −7.42213160355322890678131664661, −6.91693023974103340121958485480, −6.08516411219401110299098690591, −4.54320704668175392370062628525, −4.08162600668159128520671153046, −3.04405335082360835359014651983, −2.61979814579062458905152562597, −1.19058572886936486759117684068, 1.25772394459459126218881973657, 2.52533307895191921441656512327, 3.20059540189718734433738952548, 4.06068017359096395147018676510, 4.85364857811297964366063310576, 5.83288670562103367301706938391, 6.90175270957282161081453641918, 7.40196084390809967422998622291, 8.001643757758017533400275554669, 8.763381001254030246534307556255

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