Properties

Label 2-87120-1.1-c1-0-124
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 + 2·17-s + 2·19-s + 25-s + 6·29-s − 2·35-s − 6·37-s − 6·41-s + 2·43-s − 4·47-s − 3·49-s − 2·53-s − 8·59-s − 4·67-s + 12·71-s − 16·73-s − 2·79-s − 8·83-s − 2·85-s + 6·89-s − 2·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.755·7-s + 0.485·17-s + 0.458·19-s + 1/5·25-s + 1.11·29-s − 0.338·35-s − 0.986·37-s − 0.937·41-s + 0.304·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 1.04·59-s − 0.488·67-s + 1.42·71-s − 1.87·73-s − 0.225·79-s − 0.878·83-s − 0.216·85-s + 0.635·89-s − 0.205·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−14.14746634252477, −13.77434424639344, −13.20460862586871, −12.45825470085235, −12.26307041940894, −11.56284737589125, −11.34983570069579, −10.71830402593594, −10.08494541723699, −9.875955523960149, −8.955725500177749, −8.592915498413002, −8.163790020007114, −7.400469578883132, −7.340276804949492, −6.405270476418017, −6.017169208421624, −5.183107530259076, −4.818568592465656, −4.362002742787056, −3.387682983374988, −3.246103529624219, −2.274841505895402, −1.586159025701159, −0.9618755881788787, 0, 0.9618755881788787, 1.586159025701159, 2.274841505895402, 3.246103529624219, 3.387682983374988, 4.362002742787056, 4.818568592465656, 5.183107530259076, 6.017169208421624, 6.405270476418017, 7.340276804949492, 7.400469578883132, 8.163790020007114, 8.592915498413002, 8.955725500177749, 9.875955523960149, 10.08494541723699, 10.71830402593594, 11.34983570069579, 11.56284737589125, 12.26307041940894, 12.45825470085235, 13.20460862586871, 13.77434424639344, 14.14746634252477

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