Properties

Label 2-132600-1.1-c1-0-42
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 9-s + 4·11-s + 13-s − 17-s + 8·19-s − 27-s − 6·29-s − 8·31-s − 4·33-s − 2·37-s − 39-s − 10·41-s − 4·43-s − 7·49-s + 51-s + 6·53-s − 8·57-s − 6·61-s − 4·67-s − 8·71-s + 14·73-s + 4·79-s + 81-s + 4·83-s + 6·87-s − 14·89-s + ⋯
L(s)  = 1  − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 1.83·19-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.160·39-s − 1.56·41-s − 0.609·43-s − 49-s + 0.140·51-s + 0.824·53-s − 1.05·57-s − 0.768·61-s − 0.488·67-s − 0.949·71-s + 1.63·73-s + 0.450·79-s + 1/9·81-s + 0.439·83-s + 0.643·87-s − 1.48·89-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: \(1\)
Selberg data: \((2,\ 132600,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.74609360346515, −13.21247347740487, −12.73541952237837, −12.05432230265678, −11.83817154671082, −11.32242513537090, −10.99550550780941, −10.34652270227398, −9.753670202959326, −9.397565070451648, −8.966661154424050, −8.375494639057316, −7.696075835990368, −7.191579084301074, −6.826015602549114, −6.271423875308078, −5.606872185347889, −5.318481200403521, −4.686888822802876, −4.006020813076286, −3.449242801946443, −3.142652347741003, −1.910052120658320, −1.612096199087715, −0.8611998430805308, 0, 0.8611998430805308, 1.612096199087715, 1.910052120658320, 3.142652347741003, 3.449242801946443, 4.006020813076286, 4.686888822802876, 5.318481200403521, 5.606872185347889, 6.271423875308078, 6.826015602549114, 7.191579084301074, 7.696075835990368, 8.375494639057316, 8.966661154424050, 9.397565070451648, 9.753670202959326, 10.34652270227398, 10.99550550780941, 11.32242513537090, 11.83817154671082, 12.05432230265678, 12.73541952237837, 13.21247347740487, 13.74609360346515

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