Properties

Degree $2$
Conductor $120$
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 + 9-s − 4·11-s + 6·13-s + 15-s − 6·17-s − 4·19-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·33-s − 2·37-s + 6·39-s − 6·41-s + 12·43-s + 45-s + 8·47-s − 7·49-s − 6·51-s + 6·53-s − 4·55-s − 4·57-s + 12·59-s + 14·61-s + 6·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s + 0.960·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.840·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{120} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26949\)
\(L(\frac12)\) \(\approx\) \(1.26949\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−13.23486091060546125236692720333, −13.00221841062232627578538988126, −11.15225413084276142148114637688, −10.45550448318589263597286780431, −9.042960162718976648409639393560, −8.323057177226078640880314002796, −6.89270810769313104844169224865, −5.59585158632682227968586342616, −3.95303834850677407956821293662, −2.25002432004047770108316173263, 2.25002432004047770108316173263, 3.95303834850677407956821293662, 5.59585158632682227968586342616, 6.89270810769313104844169224865, 8.323057177226078640880314002796, 9.042960162718976648409639393560, 10.45550448318589263597286780431, 11.15225413084276142148114637688, 13.00221841062232627578538988126, 13.23486091060546125236692720333

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