Properties

Label 2-6240-1.1-c1-0-83
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 13-s − 15-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 27-s − 6·29-s − 8·31-s − 6·37-s − 39-s − 2·41-s − 4·43-s − 45-s − 7·49-s + 2·51-s + 6·53-s + 4·57-s − 2·61-s + 65-s − 8·67-s − 4·69-s + 6·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.277·13-s − 0.258·15-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s − 49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.256·61-s + 0.124·65-s − 0.977·67-s − 0.481·69-s + 0.702·73-s + 0.115·75-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: \(1\)
Selberg data: \((2,\ 6240,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
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 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.57860773190444184507718682655, −7.31410358691720893159017883814, −6.34306412903661323205670846752, −5.44511342280867385521701874991, −4.84243614399591229688061863832, −3.69094427708340770867201211666, −3.46847085138331232413249674831, −2.31242196130868732747290098696, −1.44147343949589411121227002844, 0, 1.44147343949589411121227002844, 2.31242196130868732747290098696, 3.46847085138331232413249674831, 3.69094427708340770867201211666, 4.84243614399591229688061863832, 5.44511342280867385521701874991, 6.34306412903661323205670846752, 7.31410358691720893159017883814, 7.57860773190444184507718682655

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