Properties

Label 2-87120-1.1-c1-0-117
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 + 2·17-s + 2·19-s + 6·23-s + 25-s + 2·29-s − 6·37-s + 2·41-s − 2·43-s − 2·47-s − 7·49-s − 2·53-s + 8·59-s − 4·61-s + 4·67-s − 2·71-s + 10·73-s + 4·79-s − 12·83-s − 2·85-s − 2·95-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.485·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.986·37-s + 0.312·41-s − 0.304·43-s − 0.291·47-s − 49-s − 0.274·53-s + 1.04·59-s − 0.512·61-s + 0.488·67-s − 0.237·71-s + 1.17·73-s + 0.450·79-s − 1.31·83-s − 0.216·85-s − 0.205·95-s − 1.82·97-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 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−14.11678616051947, −13.72678963514116, −13.08657335456314, −12.65532903764547, −12.20118618857810, −11.66856904005622, −11.18407198350395, −10.78033232567414, −10.16325420990684, −9.632349046284574, −9.206746214918916, −8.491719587937563, −8.191347038913660, −7.555093606625250, −6.980378823231359, −6.668479466048576, −5.884356157093315, −5.262773659159644, −4.890365431657997, −4.207737267941194, −3.503208349644700, −3.111765051234592, −2.422170530367224, −1.520117029609025, −0.9273012256259188, 0, 0.9273012256259188, 1.520117029609025, 2.422170530367224, 3.111765051234592, 3.503208349644700, 4.207737267941194, 4.890365431657997, 5.262773659159644, 5.884356157093315, 6.668479466048576, 6.980378823231359, 7.555093606625250, 8.191347038913660, 8.491719587937563, 9.206746214918916, 9.632349046284574, 10.16325420990684, 10.78033232567414, 11.18407198350395, 11.66856904005622, 12.20118618857810, 12.65532903764547, 13.08657335456314, 13.72678963514116, 14.11678616051947

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