Properties

Label 2-135240-1.1-c1-0-45
Degree $2$
Conductor $135240$
Sign $-1$
Analytic cond. $1079.89$
Root an. cond. $32.8617$
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 − 5-s + 9-s − 4·11-s + 2·13-s + 15-s − 2·17-s + 4·19-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s + 4·33-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s − 45-s − 8·47-s + 2·51-s − 10·53-s + 4·55-s − 4·57-s + 12·59-s − 14·61-s − 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s − 1.37·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(135240\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(1079.89\)
Root analytic conductor: \(32.8617\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{135240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 \)
23 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.45940478826838, −13.28140822265028, −12.60906281073700, −12.37038075907874, −11.67610664933800, −11.13487597929351, −10.97195106608322, −10.50580028407954, −9.770376040286798, −9.407023494986354, −8.898691996585409, −8.088818383610595, −7.776400349773578, −7.476889801237290, −6.621750225552850, −6.328794173555263, −5.623798017348767, −5.117510549213858, −4.819060861861065, −3.980770290000432, −3.555383241285883, −2.904852281262416, −2.204903986524376, −1.518914263968274, −0.6819592745268475, 0, 0.6819592745268475, 1.518914263968274, 2.204903986524376, 2.904852281262416, 3.555383241285883, 3.980770290000432, 4.819060861861065, 5.117510549213858, 5.623798017348767, 6.328794173555263, 6.621750225552850, 7.476889801237290, 7.776400349773578, 8.088818383610595, 8.898691996585409, 9.407023494986354, 9.770376040286798, 10.50580028407954, 10.97195106608322, 11.13487597929351, 11.67610664933800, 12.37038075907874, 12.60906281073700, 13.28140822265028, 13.45940478826838

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