Properties

Label 2-6045-1.1-c1-0-37
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 − 5·7-s + 9-s + 2·10-s + 2·11-s − 2·12-s − 13-s + 10·14-s + 15-s − 4·16-s + 6·17-s − 2·18-s + 4·19-s − 2·20-s + 5·21-s − 4·22-s + 6·23-s + 25-s + 2·26-s − 27-s − 10·28-s + 7·29-s − 2·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 1.88·7-s + 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.577·12-s − 0.277·13-s + 2.67·14-s + 0.258·15-s − 16-s + 1.45·17-s − 0.471·18-s + 0.917·19-s − 0.447·20-s + 1.09·21-s − 0.852·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 1.88·28-s + 1.29·29-s − 0.365·30-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

\(L(1)\) \(\approx\) \(0.5352589855\)
\(L(\frac12)\) \(\approx\) \(0.5352589855\)
\(L(\frac{3}{2})\) 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 \( 1 + 5 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.016915497293994256679999889326, −7.41382215511584747280755502658, −6.81607889941684992869273342186, −6.29089706733375113097579170145, −5.39299592605336602296523988742, −4.41184049419842818195815025005, −3.38592590281896754805410604343, −2.81234621044308061129173719524, −1.21452895025703685788941441892, −0.58113161202622861644870161983, 0.58113161202622861644870161983, 1.21452895025703685788941441892, 2.81234621044308061129173719524, 3.38592590281896754805410604343, 4.41184049419842818195815025005, 5.39299592605336602296523988742, 6.29089706733375113097579170145, 6.81607889941684992869273342186, 7.41382215511584747280755502658, 8.016915497293994256679999889326

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