Properties

Label 2-87120-1.1-c1-0-142
Degree $2$
Conductor $87120$
Sign $-1$
Analytic cond. $695.656$
Root an. cond. $26.3753$
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  + 5-s + 2·7-s + 4·13-s − 3·17-s − 4·19-s − 3·23-s + 25-s + 31-s + 2·35-s + 2·37-s + 6·41-s + 8·43-s + 3·47-s − 3·49-s − 9·53-s + 12·59-s − 5·61-s + 4·65-s − 2·67-s + 12·71-s − 8·73-s − 79-s − 3·85-s − 6·89-s + 8·91-s − 4·95-s − 10·97-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 1.10·13-s − 0.727·17-s − 0.917·19-s − 0.625·23-s + 1/5·25-s + 0.179·31-s + 0.338·35-s + 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.437·47-s − 3/7·49-s − 1.23·53-s + 1.56·59-s − 0.640·61-s + 0.496·65-s − 0.244·67-s + 1.42·71-s − 0.936·73-s − 0.112·79-s − 0.325·85-s − 0.635·89-s + 0.838·91-s − 0.410·95-s − 1.01·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(87120\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(695.656\)
Root analytic conductor: \(26.3753\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{87120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 87120,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
11 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−14.00191774871855, −13.83095613166058, −13.09425121895587, −12.78587983292754, −12.25294213146832, −11.48218940378188, −11.19854769993101, −10.70402553942520, −10.34734612753950, −9.531108176060180, −9.169885381804991, −8.597543879222594, −8.101414450696553, −7.763293131959178, −6.875875376134823, −6.478165409139107, −5.923748482043535, −5.470406814787289, −4.742536874433716, −4.145722750147737, −3.855124307684217, −2.798749011866780, −2.336619249139159, −1.614059186799932, −1.061297047142387, 0, 1.061297047142387, 1.614059186799932, 2.336619249139159, 2.798749011866780, 3.855124307684217, 4.145722750147737, 4.742536874433716, 5.470406814787289, 5.923748482043535, 6.478165409139107, 6.875875376134823, 7.763293131959178, 8.101414450696553, 8.597543879222594, 9.169885381804991, 9.531108176060180, 10.34734612753950, 10.70402553942520, 11.19854769993101, 11.48218940378188, 12.25294213146832, 12.78587983292754, 13.09425121895587, 13.83095613166058, 14.00191774871855

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