Properties

Label 2-87120-1.1-c1-0-101
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 − 4·7-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 25-s − 2·29-s + 8·31-s + 4·35-s + 6·37-s − 6·41-s − 8·43-s + 4·47-s + 9·49-s − 6·53-s − 4·59-s + 2·61-s − 2·65-s − 8·67-s + 6·73-s + 16·83-s − 2·85-s + 6·89-s − 8·91-s − 4·95-s − 14·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.937·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.702·73-s + 1.75·83-s − 0.216·85-s + 0.635·89-s − 0.838·91-s − 0.410·95-s − 1.42·97-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.94306879586853, −13.56163920042792, −13.25248478922044, −12.66142932875444, −12.18169884617069, −11.75587379515290, −11.25620010428714, −10.58224501342843, −10.17198285817815, −9.587976212676915, −9.277391949715715, −8.686120556285235, −8.002159168938564, −7.636673624937010, −6.920632102010427, −6.456496529355218, −6.141884399140902, −5.302304492626751, −4.903275974593733, −4.011948906461707, −3.573513389282993, −3.005677247663989, −2.651121524808239, −1.462746888519703, −0.8402885251702993, 0, 0.8402885251702993, 1.462746888519703, 2.651121524808239, 3.005677247663989, 3.573513389282993, 4.011948906461707, 4.903275974593733, 5.302304492626751, 6.141884399140902, 6.456496529355218, 6.920632102010427, 7.636673624937010, 8.002159168938564, 8.686120556285235, 9.277391949715715, 9.587976212676915, 10.17198285817815, 10.58224501342843, 11.25620010428714, 11.75587379515290, 12.18169884617069, 12.66142932875444, 13.25248478922044, 13.56163920042792, 13.94306879586853

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