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Properties

Label 2-3267-11.10-c0-0-3
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 − 0.517i·7-s − 1.93i·13-s + 16-s − 1.41i·19-s − 25-s − 0.517i·28-s − 1.73·31-s + 1.41i·43-s + 0.732·49-s − 1.93i·52-s + 1.41i·61-s + 64-s + 1.73·67-s − 1.93i·73-s + ⋯
L(s)  = 1  + 4-s − 0.517i·7-s − 1.93i·13-s + 16-s − 1.41i·19-s − 25-s − 0.517i·28-s − 1.73·31-s + 1.41i·43-s + 0.732·49-s − 1.93i·52-s + 1.41i·61-s + 64-s + 1.73·67-s − 1.93i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.536986232\)
\(L(\frac12)\) \(\approx\) \(1.536986232\)
\(L(1)\) 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 \( 1 + 0.517iT - T^{2} \)
13 \( 1 + 1.93iT - T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
79 \( 1 - 1.93iT - T^{2} \)
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

−8.581398774273914779198166497835, −7.66212898031622835293350544831, −7.42491528764338064156200076446, −6.48615943713975337362666996633, −5.71557868154761541621362927897, −5.05556988636564458177397510377, −3.82070752023145197271420709173, −3.05550009227584157588296114351, −2.22634970037347322430525235748, −0.895633476524245239422701637981, 1.80857182052585382954070619772, 2.08904261085142173650190533901, 3.48356403681026982115537413227, 4.10294859997713759860091138422, 5.38588173126502152311447486104, 5.96527130959385211439344471664, 6.75574595240329875711417235166, 7.33473840741194380848079114498, 8.148198566491282453877487706603, 8.972913020748751643222746122869

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