Properties

Degree $2$
Conductor $720$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·11-s + 6·13-s + 6·17-s + 4·19-s + 25-s + 2·29-s + 8·31-s − 2·37-s + 6·41-s − 12·43-s + 8·47-s − 7·49-s − 6·53-s + 4·55-s + 12·59-s + 14·61-s − 6·65-s − 4·67-s + 8·71-s − 6·73-s + 8·79-s − 12·83-s − 6·85-s − 10·89-s − 4·95-s + 2·97-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.20·11-s + 1.66·13-s + 1.45·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s − 49-s − 0.824·53-s + 0.539·55-s + 1.56·59-s + 1.79·61-s − 0.744·65-s − 0.488·67-s + 0.949·71-s − 0.702·73-s + 0.900·79-s − 1.31·83-s − 0.650·85-s − 1.05·89-s − 0.410·95-s + 0.203·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{720} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46588\)
\(L(\frac12)\) \(\approx\) \(1.46588\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
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

−10.38516810770871215819271893498, −9.711950421453445396906531620080, −8.364107116024692137216049760263, −8.062184279727733628805054904271, −6.97039739431553724911307537358, −5.86178029126505651814373374846, −5.06868408963538865423952877412, −3.76264729663280098898303229481, −2.89778258507562116298634418143, −1.08829539378793470967143951854, 1.08829539378793470967143951854, 2.89778258507562116298634418143, 3.76264729663280098898303229481, 5.06868408963538865423952877412, 5.86178029126505651814373374846, 6.97039739431553724911307537358, 8.062184279727733628805054904271, 8.364107116024692137216049760263, 9.711950421453445396906531620080, 10.38516810770871215819271893498

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