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Properties

Label 2-3267-11.8-c0-0-0
Degree $2$
Conductor $3267$
Sign $0.658 + 0.752i$
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  + (0.309 − 0.951i)4-s + (−0.492 − 0.159i)7-s + (1.13 + 1.56i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.304 + 0.418i)28-s + (1.40 − 1.01i)31-s + 1.41i·43-s + (−0.592 − 0.430i)49-s + (1.83 − 0.596i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + 1.73·67-s + (−1.83 − 0.596i)73-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (−0.492 − 0.159i)7-s + (1.13 + 1.56i)13-s + (−0.809 − 0.587i)16-s + (1.34 − 0.437i)19-s + (−0.309 − 0.951i)25-s + (−0.304 + 0.418i)28-s + (1.40 − 1.01i)31-s + 1.41i·43-s + (−0.592 − 0.430i)49-s + (1.83 − 0.596i)52-s + (0.831 − 1.14i)61-s + (−0.809 + 0.587i)64-s + 1.73·67-s + (−1.83 − 0.596i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 + 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.658 + 0.752i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.658 + 0.752i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.372124585\)
\(L(\frac12)\) \(\approx\) \(1.372124585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.492 + 0.159i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-1.13 - 1.56i)T + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-1.40 + 1.01i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.83 + 0.596i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.870609657124705274573867854668, −7.988096721370573416170426353331, −6.99394491256407438496958343241, −6.40739113875133326004097908022, −5.94239978384049176612817710447, −4.86030043943429285393228253724, −4.16629354733893665600468378208, −3.07971541992418771274754003250, −2.01720155902631622372864514192, −0.983078397169420318390823652212, 1.25945269566180054280318475513, 2.77689284013895541461487162776, 3.28850976420990516536381320702, 3.98833281026919273058005822233, 5.29324493533744418782176899167, 5.86752371143551787277716215240, 6.82625993105732086858842380189, 7.45824956389277756248743175799, 8.273103022521806797360665315062, 8.639985437574688295331768699888

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