Properties

Degree $2$
Conductor $4800$
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 − 4·7-s + 9-s + 2·13-s − 6·17-s − 4·19-s + 4·21-s − 27-s + 6·29-s − 8·31-s + 2·37-s − 2·39-s − 6·41-s + 4·43-s + 9·49-s + 6·51-s − 6·53-s + 4·57-s + 10·61-s − 4·63-s + 4·67-s − 2·73-s − 8·79-s + 81-s − 12·83-s − 6·87-s + 18·89-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.609·43-s + 9/7·49-s + 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.28·61-s − 0.503·63-s + 0.488·67-s − 0.234·73-s − 0.900·79-s + 1/9·81-s − 1.31·83-s − 0.643·87-s + 1.90·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4800\)    =    \(2^{6} \cdot 3 \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{4800} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4800,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7326314990\)
\(L(\frac12)\) \(\approx\) \(0.7326314990\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
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

−8.468240536036555704630848503198, −7.31357774249742150991297991120, −6.59910634542849161459156625156, −6.32472569150928839317323168550, −5.49712735041431211178430480723, −4.49368181982186146386252774551, −3.82348229275525279836763542718, −2.92788308932045914068085173884, −1.92466750882191143426507910728, −0.46487860020891297578644501903, 0.46487860020891297578644501903, 1.92466750882191143426507910728, 2.92788308932045914068085173884, 3.82348229275525279836763542718, 4.49368181982186146386252774551, 5.49712735041431211178430480723, 6.32472569150928839317323168550, 6.59910634542849161459156625156, 7.31357774249742150991297991120, 8.468240536036555704630848503198

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