Properties

Degree $2$
Conductor $348480$
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 + 4·7-s − 4·13-s − 4·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s + 4·35-s − 2·37-s + 6·41-s + 8·43-s − 6·47-s + 9·49-s − 6·53-s − 12·59-s + 2·61-s − 4·65-s + 10·67-s + 12·71-s + 16·73-s − 8·79-s − 6·89-s − 16·91-s − 4·95-s + 14·97-s + 101-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 1.10·13-s − 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 0.875·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s − 0.496·65-s + 1.22·67-s + 1.42·71-s + 1.87·73-s − 0.900·79-s − 0.635·89-s − 1.67·91-s − 0.410·95-s + 1.42·97-s + 0.0995·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(348480\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{348480} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 348480,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.609357620\)
\(L(\frac12)\) \(\approx\) \(4.609357620\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.55659688927925, −12.22034177509009, −11.40470503289055, −11.33091098159230, −10.77007105583990, −10.30835062599800, −9.916195040304457, −9.282356764997255, −8.951912274974255, −8.326736364710535, −8.024137606385770, −7.571206971591114, −7.030449482364371, −6.442185268673946, −6.156494375890461, −5.289241814871944, −5.003751026499569, −4.588674121189186, −4.274991599574984, −3.354178788714292, −2.752806442526217, −2.264000204916901, −1.844295927365790, −1.052643798591185, −0.6214821929040408, 0.6214821929040408, 1.052643798591185, 1.844295927365790, 2.264000204916901, 2.752806442526217, 3.354178788714292, 4.274991599574984, 4.588674121189186, 5.003751026499569, 5.289241814871944, 6.156494375890461, 6.442185268673946, 7.030449482364371, 7.571206971591114, 8.024137606385770, 8.326736364710535, 8.951912274974255, 9.282356764997255, 9.916195040304457, 10.30835062599800, 10.77007105583990, 11.33091098159230, 11.40470503289055, 12.22034177509009, 12.55659688927925

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