Properties

Label 2-6045-1.1-c1-0-165
Degree $2$
Conductor $6045$
Sign $1$
Analytic cond. $48.2695$
Root an. cond. $6.94763$
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  + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s + 2·7-s + 9-s + 2·10-s + 5·11-s + 2·12-s + 13-s + 4·14-s + 15-s − 4·16-s + 2·18-s + 2·19-s + 2·20-s + 2·21-s + 10·22-s − 6·23-s + 25-s + 2·26-s + 27-s + 4·28-s + 2·30-s − 31-s − 8·32-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s + 0.632·10-s + 1.50·11-s + 0.577·12-s + 0.277·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.471·18-s + 0.458·19-s + 0.447·20-s + 0.436·21-s + 2.13·22-s − 1.25·23-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + 0.365·30-s − 0.179·31-s − 1.41·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6045\)    =    \(3 \cdot 5 \cdot 13 \cdot 31\)
Sign: $1$
Analytic conductor: \(48.2695\)
Root analytic conductor: \(6.94763\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.099901334\)
\(L(\frac12)\) \(\approx\) \(7.099901334\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
13 \( 1 - T \)
31 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 9 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
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

−7.965665581046466271253371617006, −7.22686880300659616368612415181, −6.34717619551649847485405554530, −5.92693601929046982192298193608, −5.10467549272398560063239939121, −4.15518206763275895766339619575, −3.99698828503142810142025023829, −2.94319755896905148160902021703, −2.11853120521585897391620849145, −1.22976389858324984109451917995, 1.22976389858324984109451917995, 2.11853120521585897391620849145, 2.94319755896905148160902021703, 3.99698828503142810142025023829, 4.15518206763275895766339619575, 5.10467549272398560063239939121, 5.92693601929046982192298193608, 6.34717619551649847485405554530, 7.22686880300659616368612415181, 7.965665581046466271253371617006

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