Properties

Label 2-87120-1.1-c1-0-116
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 + 6·13-s − 2·17-s + 4·19-s + 25-s + 2·35-s + 6·37-s − 2·43-s + 8·47-s − 3·49-s − 2·53-s + 4·59-s − 12·61-s − 6·65-s − 4·67-s − 14·73-s − 4·79-s + 14·83-s + 2·85-s − 6·89-s − 12·91-s − 4·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 0.447·5-s − 0.755·7-s + 1.66·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.338·35-s + 0.986·37-s − 0.304·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.520·59-s − 1.53·61-s − 0.744·65-s − 0.488·67-s − 1.63·73-s − 0.450·79-s + 1.53·83-s + 0.216·85-s − 0.635·89-s − 1.25·91-s − 0.410·95-s − 0.203·97-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−14.08603993634611, −13.50715875323241, −13.21092824310420, −12.83032192989305, −11.97818509327751, −11.83850368008536, −11.06964173521534, −10.81765883586180, −10.21570712935331, −9.589492128728959, −9.091531811667131, −8.733340027128846, −8.046690269229267, −7.643680903537557, −6.957915146394925, −6.496301379832338, −5.923042543218446, −5.570583423470290, −4.621050528121175, −4.220934241657989, −3.471831485995850, −3.192334792665765, −2.446247550396804, −1.490293279264485, −0.9089457066226704, 0, 0.9089457066226704, 1.490293279264485, 2.446247550396804, 3.192334792665765, 3.471831485995850, 4.220934241657989, 4.621050528121175, 5.570583423470290, 5.923042543218446, 6.496301379832338, 6.957915146394925, 7.643680903537557, 8.046690269229267, 8.733340027128846, 9.091531811667131, 9.589492128728959, 10.21570712935331, 10.81765883586180, 11.06964173521534, 11.83850368008536, 11.97818509327751, 12.83032192989305, 13.21092824310420, 13.50715875323241, 14.08603993634611

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