Properties

Degree $2$
Conductor $346560$
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 − 2·7-s + 9-s + 2·13-s − 15-s − 2·17-s + 2·21-s − 2·23-s + 25-s − 27-s − 4·29-s + 4·31-s − 2·35-s + 2·37-s − 2·39-s − 4·41-s − 10·43-s + 45-s + 6·47-s − 3·49-s + 2·51-s + 6·53-s + 4·59-s + 10·61-s − 2·63-s + 2·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.624·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.251·63-s + 0.248·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(346560\)    =    \(2^{6} \cdot 3 \cdot 5 \cdot 19^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{346560} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 346560,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.331363838\)
\(L(\frac12)\) \(\approx\) \(1.331363838\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
19 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−12.57047928672025, −12.03394653529374, −11.70882735918894, −11.16642067308246, −10.79275254244405, −10.19469763357439, −9.905592653546170, −9.526601570205239, −8.899479058297178, −8.520501529268649, −7.991972492860887, −7.397236211989159, −6.729717836414781, −6.578407999504500, −6.112636403069989, −5.476601049550071, −5.221664434492522, −4.540576318087220, −3.891021107310660, −3.611493403854203, −2.852067385008273, −2.308607093376422, −1.707303415948011, −1.040418711367283, −0.3417791676628779, 0.3417791676628779, 1.040418711367283, 1.707303415948011, 2.308607093376422, 2.852067385008273, 3.611493403854203, 3.891021107310660, 4.540576318087220, 5.221664434492522, 5.476601049550071, 6.112636403069989, 6.578407999504500, 6.729717836414781, 7.397236211989159, 7.991972492860887, 8.520501529268649, 8.899479058297178, 9.526601570205239, 9.905592653546170, 10.19469763357439, 10.79275254244405, 11.16642067308246, 11.70882735918894, 12.03394653529374, 12.57047928672025

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