Properties

Label 2-7440-1.1-c1-0-49
Degree $2$
Conductor $7440$
Sign $1$
Analytic cond. $59.4086$
Root an. cond. $7.70770$
Motivic weight $1$
Arithmetic yes
Rational yes
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 + 4·11-s − 4·13-s − 15-s + 2·17-s + 8·19-s − 2·21-s + 8·23-s + 25-s − 27-s + 4·29-s + 31-s − 4·33-s + 2·35-s − 12·37-s + 4·39-s + 10·41-s − 8·43-s + 45-s + 4·47-s − 3·49-s − 2·51-s + 6·53-s + 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s + 0.179·31-s − 0.696·33-s + 0.338·35-s − 1.97·37-s + 0.640·39-s + 1.56·41-s − 1.21·43-s + 0.149·45-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 0.824·53-s + 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7440\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(59.4086\)
Root analytic conductor: \(7.70770\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7440,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.483967128\)
\(L(\frac12)\) \(\approx\) \(2.483967128\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
31 \( 1 - T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
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

−7.73607329399803119395495322518, −7.04545572599436585427854318592, −6.69974762581254571811671183419, −5.53697114966168480167946148377, −5.21425268691975651513929567957, −4.55910921470387361475142056831, −3.54592134070390671333408463172, −2.69356100104222580603901645737, −1.52843138728323757710674036806, −0.917309887421169471303075548523, 0.917309887421169471303075548523, 1.52843138728323757710674036806, 2.69356100104222580603901645737, 3.54592134070390671333408463172, 4.55910921470387361475142056831, 5.21425268691975651513929567957, 5.53697114966168480167946148377, 6.69974762581254571811671183419, 7.04545572599436585427854318592, 7.73607329399803119395495322518

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