Properties

Label 2-87120-1.1-c1-0-130
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 − 2·13-s + 2·19-s + 25-s − 8·31-s + 2·35-s + 2·37-s + 2·43-s − 3·49-s − 6·53-s − 12·59-s − 2·61-s − 2·65-s + 4·67-s − 2·73-s − 10·79-s + 12·83-s + 6·89-s − 4·91-s + 2·95-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s − 0.554·13-s + 0.458·19-s + 1/5·25-s − 1.43·31-s + 0.338·35-s + 0.328·37-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 1.56·59-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.234·73-s − 1.12·79-s + 1.31·83-s + 0.635·89-s − 0.419·91-s + 0.205·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−14.28831912891513, −13.75744228667649, −13.08119032739852, −12.74076694330598, −12.20125118238276, −11.60491725564650, −11.21954179951708, −10.68423387672709, −10.21889264437192, −9.580070592224168, −9.198013657772345, −8.716774584488326, −7.981733900548898, −7.585218528784864, −7.171719101217929, −6.387069634088433, −5.944246143351336, −5.290113826286201, −4.853621542267319, −4.366595132219289, −3.523646201411583, −3.018778447035390, −2.168791050923002, −1.752989403547708, −0.9880444543774239, 0, 0.9880444543774239, 1.752989403547708, 2.168791050923002, 3.018778447035390, 3.523646201411583, 4.366595132219289, 4.853621542267319, 5.290113826286201, 5.944246143351336, 6.387069634088433, 7.171719101217929, 7.585218528784864, 7.981733900548898, 8.716774584488326, 9.198013657772345, 9.580070592224168, 10.21889264437192, 10.68423387672709, 11.21954179951708, 11.60491725564650, 12.20125118238276, 12.74076694330598, 13.08119032739852, 13.75744228667649, 14.28831912891513

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