Properties

Label 2-87120-1.1-c1-0-111
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 − 7-s − 2·13-s + 2·19-s + 25-s − 6·29-s + 4·31-s − 35-s − 4·37-s + 9·41-s − 43-s − 3·47-s − 6·49-s + 6·53-s + 61-s − 2·65-s + 13·67-s − 12·71-s + 16·73-s − 10·79-s − 12·83-s + 3·89-s + 2·91-s + 2·95-s − 10·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.554·13-s + 0.458·19-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s − 0.657·37-s + 1.40·41-s − 0.152·43-s − 0.437·47-s − 6/7·49-s + 0.824·53-s + 0.128·61-s − 0.248·65-s + 1.58·67-s − 1.42·71-s + 1.87·73-s − 1.12·79-s − 1.31·83-s + 0.317·89-s + 0.209·91-s + 0.205·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-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 \( 1 + T + p T^{2} \)
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 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−14.10899950912650, −13.66960336756105, −13.11599973781356, −12.74703215646351, −12.22762156325579, −11.70413689589792, −11.15525389969186, −10.69831501831522, −10.04703407640394, −9.605709429576414, −9.371013789055391, −8.621668207632298, −8.116160308863210, −7.542097497600143, −6.921126994723563, −6.626341471688847, −5.716852531154896, −5.580808092854451, −4.810463014100404, −4.240416783264914, −3.562810658072527, −2.947504761343161, −2.361588688439998, −1.695234784158665, −0.8912516605150102, 0, 0.8912516605150102, 1.695234784158665, 2.361588688439998, 2.947504761343161, 3.562810658072527, 4.240416783264914, 4.810463014100404, 5.580808092854451, 5.716852531154896, 6.626341471688847, 6.921126994723563, 7.542097497600143, 8.116160308863210, 8.621668207632298, 9.371013789055391, 9.605709429576414, 10.04703407640394, 10.69831501831522, 11.15525389969186, 11.70413689589792, 12.22762156325579, 12.74703215646351, 13.11599973781356, 13.66960336756105, 14.10899950912650

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