Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 19 \cdot 53 $
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 + 3-s + 4-s + 2·5-s + 6-s + 4·7-s + 8-s + 9-s + 2·10-s + 12-s + 4·13-s + 4·14-s + 2·15-s + 16-s − 6·17-s + 18-s − 19-s + 2·20-s + 4·21-s + 24-s − 25-s + 4·26-s + 27-s + 4·28-s + 6·29-s + 2·30-s + 4·31-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s + 1.10·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s + 0.872·21-s + 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.365·30-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6042\)    =    \(2 \cdot 3 \cdot 19 \cdot 53\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6042} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6042,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(6.079061319\)
\(L(\frac12)\)  \(\approx\)  \(6.079061319\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;19,\;53\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;19,\;53\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
19 \( 1 + T \)
53 \( 1 - T \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.140290228630297279944580099510, −7.38787382526592493864916637260, −6.37155307349173285287549425146, −6.08418025981844902591785628570, −4.92731636918462634620034174463, −4.60923037696084761578448190559, −3.73565787993472093859065600673, −2.67057526604608401220782403836, −1.95507177782677035128785949567, −1.30705872212461425047618725823, 1.30705872212461425047618725823, 1.95507177782677035128785949567, 2.67057526604608401220782403836, 3.73565787993472093859065600673, 4.60923037696084761578448190559, 4.92731636918462634620034174463, 6.08418025981844902591785628570, 6.37155307349173285287549425146, 7.38787382526592493864916637260, 8.140290228630297279944580099510

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