Properties

Label 2-3300-1.1-c1-0-8
Degree $2$
Conductor $3300$
Sign $1$
Analytic cond. $26.3506$
Root an. cond. $5.13328$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 7-s + 9-s − 11-s + 4·13-s − 3·17-s + 5·19-s − 21-s − 3·23-s − 27-s − 6·29-s + 8·31-s + 33-s + 7·37-s − 4·39-s + 9·41-s − 8·43-s + 3·47-s − 6·49-s + 3·51-s − 6·53-s − 5·57-s + 3·59-s + 14·61-s + 63-s − 2·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.727·17-s + 1.14·19-s − 0.218·21-s − 0.625·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.174·33-s + 1.15·37-s − 0.640·39-s + 1.40·41-s − 1.21·43-s + 0.437·47-s − 6/7·49-s + 0.420·51-s − 0.824·53-s − 0.662·57-s + 0.390·59-s + 1.79·61-s + 0.125·63-s − 0.244·67-s + 0.361·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3300\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(26.3506\)
Root analytic conductor: \(5.13328\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3300} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3300,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.605408394\)
\(L(\frac12)\) \(\approx\) \(1.605408394\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 \)
11 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 11 T + p T^{2} \)
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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

−8.533366363768036665894131485740, −7.907399722356293446239967589922, −7.14222300682144100020080834629, −6.22559717910638491456369263143, −5.72795927912524235626537501289, −4.80227563736251391425127187622, −4.09061939154891115410203564799, −3.08417434889935858016967485118, −1.89583451694284492613556111631, −0.802045474788085516711551537887, 0.802045474788085516711551537887, 1.89583451694284492613556111631, 3.08417434889935858016967485118, 4.09061939154891115410203564799, 4.80227563736251391425127187622, 5.72795927912524235626537501289, 6.22559717910638491456369263143, 7.14222300682144100020080834629, 7.907399722356293446239967589922, 8.533366363768036665894131485740

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