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Properties

Label 2-3267-11.10-c0-0-2
Degree $2$
Conductor $3267$
Sign $0.522 + 0.852i$
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  + 4-s − 1.93i·7-s − 0.517i·13-s + 16-s + 1.41i·19-s − 25-s − 1.93i·28-s + 1.73·31-s − 1.41i·43-s − 2.73·49-s − 0.517i·52-s − 1.41i·61-s + 64-s − 1.73·67-s − 0.517i·73-s + ⋯
L(s)  = 1  + 4-s − 1.93i·7-s − 0.517i·13-s + 16-s + 1.41i·19-s − 25-s − 1.93i·28-s + 1.73·31-s − 1.41i·43-s − 2.73·49-s − 0.517i·52-s − 1.41i·61-s + 64-s − 1.73·67-s − 0.517i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3267\)    =    \(3^{3} \cdot 11^{2}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(1.63044\)
Root analytic conductor: \(1.27688\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3267} (2782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3267,\ (\ :0),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.601430125\)
\(L(\frac12)\) \(\approx\) \(1.601430125\)
\(L(1)\) 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 \( 1 - T^{2} \)
5 \( 1 + T^{2} \)
7 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + 0.517iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.73T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + 0.517iT - T^{2} \)
79 \( 1 - 0.517iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.412970529283329585347392592814, −7.71506124318885082241445901818, −7.38100288563971429712876300100, −6.47896242437598672541613328868, −5.93224957270222222737158279360, −4.78494384024374958637999385871, −3.83425013407645239092474736472, −3.29401099533779911087651522055, −1.99624627511606155583601100987, −0.987091595782885604660525575453, 1.62842713776413164374217909733, 2.58563405145722755519026087337, 2.94873373982252916440742829324, 4.41526534156070664886655197547, 5.29381553457790105101882656502, 6.10000461034407803781073997851, 6.48780612808044916235831420806, 7.45777253098177644946224897216, 8.249450495780411953343866354730, 8.929265338401141616198084325052

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