Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 67 $
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 + 2.08·7-s + 8-s + 10-s − 2.20·11-s − 4.28·13-s + 2.08·14-s + 16-s + 4.93·17-s + 4.93·19-s + 20-s − 2.20·22-s + 3.35·23-s + 25-s − 4.28·26-s + 2.08·28-s + 2.81·29-s − 2.28·31-s + 32-s + 4.93·34-s + 2.08·35-s − 0.727·37-s + 4.93·38-s + 40-s + 1.47·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.787·7-s + 0.353·8-s + 0.316·10-s − 0.664·11-s − 1.18·13-s + 0.556·14-s + 0.250·16-s + 1.19·17-s + 1.13·19-s + 0.223·20-s − 0.470·22-s + 0.699·23-s + 0.200·25-s − 0.840·26-s + 0.393·28-s + 0.521·29-s − 0.410·31-s + 0.176·32-s + 0.845·34-s + 0.352·35-s − 0.119·37-s + 0.800·38-s + 0.158·40-s + 0.230·41-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

\( d \)  =  \(2\)
\( N \)  =  \(6030\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6030} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6030,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(3.943946023\)
\(L(\frac12)\)  \(\approx\)  \(3.943946023\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;67\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
5 \( 1 - T \)
67 \( 1 + T \)
good7 \( 1 - 2.08T + 7T^{2} \)
11 \( 1 + 2.20T + 11T^{2} \)
13 \( 1 + 4.28T + 13T^{2} \)
17 \( 1 - 4.93T + 17T^{2} \)
19 \( 1 - 4.93T + 19T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 + 2.28T + 31T^{2} \)
37 \( 1 + 0.727T + 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 - 4.41T + 43T^{2} \)
47 \( 1 + 9.38T + 47T^{2} \)
53 \( 1 - 6.16T + 53T^{2} \)
59 \( 1 - 1.47T + 59T^{2} \)
61 \( 1 - 4.08T + 61T^{2} \)
71 \( 1 - 4.84T + 71T^{2} \)
73 \( 1 + 0.544T + 73T^{2} \)
79 \( 1 - 4.83T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 2.18T + 89T^{2} \)
97 \( 1 - 2.20T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.72633626760727591116733084234, −7.51678074859552183376989279629, −6.62131211810840599409297430544, −5.66844285181768270962852543711, −5.10104356570960563610654813863, −4.78954313608282756113983000815, −3.56751063087084859015278106674, −2.83083279810865154989203454986, −2.03177459569741288596429218517, −0.973024210035254159172176342806, 0.973024210035254159172176342806, 2.03177459569741288596429218517, 2.83083279810865154989203454986, 3.56751063087084859015278106674, 4.78954313608282756113983000815, 5.10104356570960563610654813863, 5.66844285181768270962852543711, 6.62131211810840599409297430544, 7.51678074859552183376989279629, 7.72633626760727591116733084234

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