Properties

Label 2-6240-1.1-c1-0-40
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 + 7-s + 9-s − 3·11-s − 13-s + 15-s + 3·17-s + 6·19-s + 21-s + 23-s + 25-s + 27-s + 4·29-s + 8·31-s − 3·33-s + 35-s − 3·37-s − 39-s − 3·41-s + 6·43-s + 45-s − 12·47-s − 6·49-s + 3·51-s − 53-s − 3·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s + 0.727·17-s + 1.37·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s − 0.522·33-s + 0.169·35-s − 0.493·37-s − 0.160·39-s − 0.468·41-s + 0.914·43-s + 0.149·45-s − 1.75·47-s − 6/7·49-s + 0.420·51-s − 0.137·53-s − 0.404·55-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.996948797\)
\(L(\frac12)\) \(\approx\) \(2.996948797\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 15 T + p T^{2} \)
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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

−8.047302069950730281574356946876, −7.49265990556339247265918395954, −6.74053198957974933118875394690, −5.84530554752435688182227970096, −5.08370319766216404367091124283, −4.61248178209170083285750463390, −3.33282812219536295115237024910, −2.87376314889479207438380090675, −1.90254529231889943391348065422, −0.908515222001293130349717551836, 0.908515222001293130349717551836, 1.90254529231889943391348065422, 2.87376314889479207438380090675, 3.33282812219536295115237024910, 4.61248178209170083285750463390, 5.08370319766216404367091124283, 5.84530554752435688182227970096, 6.74053198957974933118875394690, 7.49265990556339247265918395954, 8.047302069950730281574356946876

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