Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 13 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 5-s + 6-s + 8-s + 9-s − 10-s + 11-s + 12-s − 13-s − 15-s + 16-s − 3·17-s + 18-s − 19-s − 20-s + 22-s − 4·23-s + 24-s − 4·25-s − 26-s + 27-s + 9·29-s − 30-s − 10·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 1.67·29-s − 0.182·30-s − 1.79·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 42042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(42042\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{42042} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 42042,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;11,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;11,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 \)
11 \( 1 - T \)
13 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.94163237915249, −14.45900202622894, −13.97150116655796, −13.48282197798927, −12.99343324479501, −12.40399626062719, −12.00588174429104, −11.42501403849383, −10.91138671716544, −10.36074176866223, −9.617677096838252, −9.295754809267772, −8.428442239081648, −8.078803647467137, −7.489799671126971, −6.882340470201658, −6.358769768725279, −5.755153215884486, −4.994962366673497, −4.348337916235738, −3.951895847214428, −3.363215128339387, −2.507929139811736, −2.104048064549758, −1.153303624199647, 0, 1.153303624199647, 2.104048064549758, 2.507929139811736, 3.363215128339387, 3.951895847214428, 4.348337916235738, 4.994962366673497, 5.755153215884486, 6.358769768725279, 6.882340470201658, 7.489799671126971, 8.078803647467137, 8.428442239081648, 9.295754809267772, 9.617677096838252, 10.36074176866223, 10.91138671716544, 11.42501403849383, 12.00588174429104, 12.40399626062719, 12.99343324479501, 13.48282197798927, 13.97150116655796, 14.45900202622894, 14.94163237915249

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