Properties

Label 2-6240-1.1-c1-0-8
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 − 2·7-s + 9-s − 13-s + 15-s + 6·19-s + 2·21-s + 2·23-s + 25-s − 27-s − 4·29-s − 8·31-s + 2·35-s − 10·37-s + 39-s − 2·41-s + 12·43-s − 45-s + 4·47-s − 3·49-s + 6·53-s − 6·57-s − 8·59-s − 2·61-s − 2·63-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s + 1.37·19-s + 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s − 1.43·31-s + 0.338·35-s − 1.64·37-s + 0.160·39-s − 0.312·41-s + 1.82·43-s − 0.149·45-s + 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.794·57-s − 1.04·59-s − 0.256·61-s − 0.251·63-s + 0.124·65-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

\(L(1)\) \(\approx\) \(0.9477548306\)
\(L(\frac12)\) \(\approx\) \(0.9477548306\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−7.84159260764064842968865159418, −7.19773607781556682912964485518, −6.83613644301337970943682406526, −5.69669129713303018125356379802, −5.42491641941761392565163766594, −4.40954908194062381260735833941, −3.60513748978311367021979424100, −2.95584723672102699188506946539, −1.71348486018265623815718068995, −0.52144196617683708403193604902, 0.52144196617683708403193604902, 1.71348486018265623815718068995, 2.95584723672102699188506946539, 3.60513748978311367021979424100, 4.40954908194062381260735833941, 5.42491641941761392565163766594, 5.69669129713303018125356379802, 6.83613644301337970943682406526, 7.19773607781556682912964485518, 7.84159260764064842968865159418

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