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Properties

Label 2-3267-11.8-c0-0-3
Degree $2$
Conductor $3267$
Sign $-0.969 - 0.245i$
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 + (−1.83 − 0.596i)7-s + (0.304 + 0.418i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (−1.13 + 1.56i)28-s + (−1.40 + 1.01i)31-s − 1.41i·43-s + (2.21 + 1.60i)49-s + (0.492 − 0.159i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (−0.492 − 0.159i)73-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)4-s + (−1.83 − 0.596i)7-s + (0.304 + 0.418i)13-s + (−0.809 − 0.587i)16-s + (−1.34 + 0.437i)19-s + (−0.309 − 0.951i)25-s + (−1.13 + 1.56i)28-s + (−1.40 + 1.01i)31-s − 1.41i·43-s + (2.21 + 1.60i)49-s + (0.492 − 0.159i)52-s + (−0.831 + 1.14i)61-s + (−0.809 + 0.587i)64-s − 1.73·67-s + (−0.492 − 0.159i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $-0.969 - 0.245i$
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: \((2,\ 3267,\ (\ :0),\ -0.969 - 0.245i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2814035339\)
\(L(\frac12)\) \(\approx\) \(0.2814035339\)
\(L(1)\) 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 \( 1 + (1.83 + 0.596i)T + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.304 - 0.418i)T + (-0.309 + 0.951i)T^{2} \)
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 \( 1 + (0.492 + 0.159i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2} \)
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

−8.709741581406830751127542904935, −7.36890440183870921641968434093, −6.81208981858364561726121642537, −6.19530651296143474492933288812, −5.68926862580309376658556635020, −4.43987910172009155018175336989, −3.72729047365320498732631439077, −2.73158617323826797637217215702, −1.65383512614875707759921920346, −0.15258689065323235842523057926, 2.09112482333349634024479048526, 2.98738465729378153963531781143, 3.52747787912402605060295333950, 4.39645062846893636768718656376, 5.74198513557267096752641050781, 6.25632512553464191221870340413, 6.97182912173584667892716283589, 7.67782804496646100838645389234, 8.576521189996604884775192467180, 9.179963402800902264464744338325

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