Properties

Label 2-152880-1.1-c1-0-208
Degree $2$
Conductor $152880$
Sign $1$
Analytic cond. $1220.75$
Root an. cond. $34.9392$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s + 9-s − 4·11-s − 13-s + 15-s − 6·17-s + 4·23-s + 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 2·37-s + 39-s − 10·41-s + 4·43-s − 45-s + 8·47-s + 6·51-s − 2·53-s + 4·55-s + 4·59-s − 14·61-s + 65-s + 12·67-s − 4·69-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.834·23-s + 1/5·25-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 − 0.149·45-s + 1.16·47-s + 0.840·51-s − 0.274·53-s + 0.539·55-s + 0.520·59-s − 1.79·61-s + 0.124·65-s + 1.46·67-s − 0.481·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152880\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1220.75\)
Root analytic conductor: \(34.9392\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{152880} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 152880,\ (\ :1/2),\ 1)\)

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 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−13.75942898498718, −13.12689539340498, −12.85397157047298, −12.45155686211319, −11.90717560119331, −11.19836511900083, −10.95674431726671, −10.75620950409082, −10.01565647236243, −9.501072286302465, −8.965533103275913, −8.520766872931369, −7.923101323748733, −7.291752855230633, −7.122723626366429, −6.489050040681294, −5.849110381999586, −5.176078290364540, −5.074378901821433, −4.323304225529587, −3.767582867152488, −3.194148433811596, −2.373997405590718, −2.013612476502947, −1.085401129577982, 0, 0, 1.085401129577982, 2.013612476502947, 2.373997405590718, 3.194148433811596, 3.767582867152488, 4.323304225529587, 5.074378901821433, 5.176078290364540, 5.849110381999586, 6.489050040681294, 7.122723626366429, 7.291752855230633, 7.923101323748733, 8.520766872931369, 8.965533103275913, 9.501072286302465, 10.01565647236243, 10.75620950409082, 10.95674431726671, 11.19836511900083, 11.90717560119331, 12.45155686211319, 12.85397157047298, 13.12689539340498, 13.75942898498718

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