Properties

Label 2-6240-1.1-c1-0-48
Degree $2$
Conductor $6240$
Sign $1$
Analytic cond. $49.8266$
Root an. cond. $7.05879$
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 + 4·7-s + 9-s + 13-s − 15-s + 2·17-s + 8·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 4·35-s + 6·37-s − 39-s − 6·41-s − 4·43-s + 45-s + 4·47-s + 9·49-s − 2·51-s + 6·53-s − 8·57-s + 8·59-s − 10·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s + 0.485·17-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s − 0.160·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s − 1.05·57-s + 1.04·59-s − 1.28·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6240\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(49.8266\)
Root analytic conductor: \(7.05879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{6240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623222105\)
\(L(\frac12)\) \(\approx\) \(2.623222105\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
good7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.944490996203046871006740236534, −7.37195814834809496724933212125, −6.68191318195853732161710038721, −5.63118185057842449000250085630, −5.28635645982431779207741724983, −4.68425346732578023214280853595, −3.69498404820957952992327762228, −2.70189473146539199878359027873, −1.56097181112985290639590099657, −0.991953726160196555799716449956, 0.991953726160196555799716449956, 1.56097181112985290639590099657, 2.70189473146539199878359027873, 3.69498404820957952992327762228, 4.68425346732578023214280853595, 5.28635645982431779207741724983, 5.63118185057842449000250085630, 6.68191318195853732161710038721, 7.37195814834809496724933212125, 7.944490996203046871006740236534

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