Properties

Degree $2$
Conductor $6030$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 5-s + 4.33·7-s + 8-s + 10-s + 4.74·11-s + 0.409·13-s + 4.33·14-s + 16-s − 5.77·17-s − 5.77·19-s + 20-s + 4.74·22-s + 9.36·23-s + 25-s + 0.409·26-s + 4.33·28-s + 1.30·29-s + 2.40·31-s + 32-s − 5.77·34-s + 4.33·35-s + 3.02·37-s − 5.77·38-s + 40-s − 1.71·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.63·7-s + 0.353·8-s + 0.316·10-s + 1.43·11-s + 0.113·13-s + 1.15·14-s + 0.250·16-s − 1.40·17-s − 1.32·19-s + 0.223·20-s + 1.01·22-s + 1.95·23-s + 0.200·25-s + 0.0802·26-s + 0.819·28-s + 0.242·29-s + 0.432·31-s + 0.176·32-s − 0.990·34-s + 0.732·35-s + 0.497·37-s − 0.936·38-s + 0.158·40-s − 0.268·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{6030} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6030,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.856914599\)
\(L(\frac12)\) \(\approx\) \(4.856914599\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
67 \( 1 + T \)
good7 \( 1 - 4.33T + 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 0.409T + 13T^{2} \)
17 \( 1 + 5.77T + 17T^{2} \)
19 \( 1 + 5.77T + 19T^{2} \)
23 \( 1 - 9.36T + 23T^{2} \)
29 \( 1 - 1.30T + 29T^{2} \)
31 \( 1 - 2.40T + 31T^{2} \)
37 \( 1 - 3.02T + 37T^{2} \)
41 \( 1 + 1.71T + 41T^{2} \)
43 \( 1 + 9.49T + 43T^{2} \)
47 \( 1 - 1.51T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 1.71T + 59T^{2} \)
61 \( 1 - 6.33T + 61T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + 8.05T + 73T^{2} \)
79 \( 1 - 7.64T + 79T^{2} \)
83 \( 1 + 8.57T + 83T^{2} \)
89 \( 1 - 9.08T + 89T^{2} \)
97 \( 1 + 4.74T + 97T^{2} \)
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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

−8.198669804035354407981298936259, −6.98924109777514627592385074383, −6.75788608481903978949861074201, −5.90055250539786635375712053750, −4.99715441297207004980174738992, −4.52128669086471931970452027593, −3.92593429429863213739557066390, −2.69559758127604620805874421086, −1.88089965247911845607076969805, −1.16785206999067924713004166765, 1.16785206999067924713004166765, 1.88089965247911845607076969805, 2.69559758127604620805874421086, 3.92593429429863213739557066390, 4.52128669086471931970452027593, 4.99715441297207004980174738992, 5.90055250539786635375712053750, 6.75788608481903978949861074201, 6.98924109777514627592385074383, 8.198669804035354407981298936259

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