Properties

Degree $2$
Conductor $69360$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 5-s − 4·7-s + 9-s + 2·13-s + 15-s + 4·19-s − 4·21-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·35-s − 2·37-s + 2·39-s + 6·41-s + 4·43-s + 45-s + 9·49-s − 6·53-s + 4·57-s + 10·61-s − 4·63-s + 2·65-s + 4·67-s − 2·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.503·63-s + 0.248·65-s + 0.488·67-s − 0.234·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69360\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 17^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{69360} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 69360,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.371416257\)
\(L(\frac12)\) \(\approx\) \(3.371416257\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
17 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−14.11433590456015, −13.62648687873405, −13.22958291556172, −12.73226661873990, −12.31451290738799, −11.71851441685929, −11.10551514674242, −10.37716167882628, −9.987195619216976, −9.683351798503603, −9.030626601706804, −8.710860171281177, −7.992734008371133, −7.455603481152782, −6.796456497374912, −6.321901734253546, −6.000727662182286, −5.190952857628411, −4.570014322638443, −3.800596636511722, −3.320517916957984, −2.768372300803544, −2.280429332860410, −1.213639792480949, −0.6472613868858427, 0.6472613868858427, 1.213639792480949, 2.280429332860410, 2.768372300803544, 3.320517916957984, 3.800596636511722, 4.570014322638443, 5.190952857628411, 6.000727662182286, 6.321901734253546, 6.796456497374912, 7.455603481152782, 7.992734008371133, 8.710860171281177, 9.030626601706804, 9.683351798503603, 9.987195619216976, 10.37716167882628, 11.10551514674242, 11.71851441685929, 12.31451290738799, 12.73226661873990, 13.22958291556172, 13.62648687873405, 14.11433590456015

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