Properties

Label 2-87120-1.1-c1-0-129
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 + 25-s − 2·29-s − 4·35-s − 2·37-s − 6·41-s + 8·47-s + 9·49-s + 2·53-s + 4·59-s + 10·61-s + 2·65-s − 4·67-s − 14·73-s + 16·79-s + 2·85-s − 10·89-s − 8·91-s − 4·95-s − 6·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s + 1.51·7-s − 0.554·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.371·29-s − 0.676·35-s − 0.328·37-s − 0.937·41-s + 1.16·47-s + 9/7·49-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 1.63·73-s + 1.80·79-s + 0.216·85-s − 1.05·89-s − 0.838·91-s − 0.410·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−14.21804173053980, −13.63143239632803, −13.34874620182990, −12.50331395752878, −12.06525364722659, −11.69002556884623, −11.25477806930473, −10.75187700984921, −10.30424256806622, −9.598409372592782, −9.133002383643487, −8.412615885639694, −8.219949070373052, −7.532817442917649, −7.163631419606343, −6.669536540514605, −5.698227358550618, −5.314421411729053, −4.845154046548255, −4.221055534015807, −3.775558970943755, −2.909506602527096, −2.306211389101198, −1.613420235468210, −0.9839425373752766, 0, 0.9839425373752766, 1.613420235468210, 2.306211389101198, 2.909506602527096, 3.775558970943755, 4.221055534015807, 4.845154046548255, 5.314421411729053, 5.698227358550618, 6.669536540514605, 7.163631419606343, 7.532817442917649, 8.219949070373052, 8.412615885639694, 9.133002383643487, 9.598409372592782, 10.30424256806622, 10.75187700984921, 11.25477806930473, 11.69002556884623, 12.06525364722659, 12.50331395752878, 13.34874620182990, 13.63143239632803, 14.21804173053980

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