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Properties

Label 2-135240-1.1-c1-0-36
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 4·13-s − 15-s + 17-s + 23-s + 25-s + 27-s − 7·29-s + 7·31-s − 3·37-s + 4·39-s + 5·41-s + 12·43-s − 45-s + 6·47-s + 51-s + 3·53-s − 7·59-s − 2·61-s − 4·65-s − 3·67-s + 69-s − 3·71-s + 4·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.10·13-s − 0.258·15-s + 0.242·17-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 1.29·29-s + 1.25·31-s − 0.493·37-s + 0.640·39-s + 0.780·41-s + 1.82·43-s − 0.149·45-s + 0.875·47-s + 0.140·51-s + 0.412·53-s − 0.911·59-s − 0.256·61-s − 0.496·65-s − 0.366·67-s + 0.120·69-s − 0.356·71-s + 0.468·73-s + 0.115·75-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 135240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.490201551\)
\(L(\frac12)\) \(\approx\) \(3.490201551\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 7 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−13.38667394914289, −13.04974649929256, −12.54847352931524, −12.02005796344974, −11.53453101504548, −10.98068650490546, −10.63304918735525, −10.08753214741709, −9.462806131604917, −8.890615032035101, −8.753939611481875, −8.019206306256197, −7.547489164244273, −7.274339451270462, −6.477004185179601, −5.981842112248713, −5.549008593968216, −4.727897659539369, −4.194096892194742, −3.790562441905512, −3.169310518503834, −2.653882236797647, −1.922587379476299, −1.197650947087237, −0.5857756465577426, 0.5857756465577426, 1.197650947087237, 1.922587379476299, 2.653882236797647, 3.169310518503834, 3.790562441905512, 4.194096892194742, 4.727897659539369, 5.549008593968216, 5.981842112248713, 6.477004185179601, 7.274339451270462, 7.547489164244273, 8.019206306256197, 8.753939611481875, 8.890615032035101, 9.462806131604917, 10.08753214741709, 10.63304918735525, 10.98068650490546, 11.53453101504548, 12.02005796344974, 12.54847352931524, 13.04974649929256, 13.38667394914289

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