Properties

Label 2-132600-1.1-c1-0-13
Degree $2$
Conductor $132600$
Sign $1$
Analytic cond. $1058.81$
Root an. cond. $32.5394$
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 + 4·7-s + 9-s − 13-s − 17-s − 4·21-s − 27-s − 2·29-s + 4·31-s + 2·37-s + 39-s − 6·41-s − 4·43-s + 12·47-s + 9·49-s + 51-s − 6·53-s − 8·59-s + 14·61-s + 4·63-s − 8·67-s − 12·71-s − 2·73-s − 8·79-s + 81-s + 2·87-s + 2·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.242·17-s − 0.872·21-s − 0.192·27-s − 0.371·29-s + 0.718·31-s + 0.328·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.140·51-s − 0.824·53-s − 1.04·59-s + 1.79·61-s + 0.503·63-s − 0.977·67-s − 1.42·71-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 0.214·87-s + 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132600\)    =    \(2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(1058.81\)
Root analytic conductor: \(32.5394\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{132600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 132600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.008425359\)
\(L(\frac12)\) \(\approx\) \(2.008425359\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
17 \( 1 + T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.43115018808204, −13.04976424418134, −12.32124437596413, −11.86702296050081, −11.68383977171340, −10.99745748946830, −10.74351310608064, −10.20089401354478, −9.647617964404675, −9.062143780975545, −8.464796240042100, −8.130543153264818, −7.515703333057797, −7.115708595621151, −6.527926310653486, −5.866214258146388, −5.398810652119740, −4.940895714594958, −4.358075580891267, −4.070751245243086, −3.099240087962348, −2.482832981187390, −1.717936218701735, −1.334648437355791, −0.4507247207757432, 0.4507247207757432, 1.334648437355791, 1.717936218701735, 2.482832981187390, 3.099240087962348, 4.070751245243086, 4.358075580891267, 4.940895714594958, 5.398810652119740, 5.866214258146388, 6.527926310653486, 7.115708595621151, 7.515703333057797, 8.130543153264818, 8.464796240042100, 9.062143780975545, 9.647617964404675, 10.20089401354478, 10.74351310608064, 10.99745748946830, 11.68383977171340, 11.86702296050081, 12.32124437596413, 13.04976424418134, 13.43115018808204

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