Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2} $
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 + 4-s + 5-s + 2·7-s − 8-s − 10-s − 13-s − 2·14-s + 16-s + 2·19-s + 20-s − 8·23-s + 25-s + 26-s + 2·28-s + 6·29-s + 4·31-s − 32-s + 2·35-s − 2·37-s − 2·38-s − 40-s + 6·41-s − 2·43-s + 8·46-s − 3·49-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.458·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.338·35-s − 0.328·37-s − 0.324·38-s − 0.158·40-s + 0.937·41-s − 0.304·43-s + 1.17·46-s − 3/7·49-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{338130} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 338130,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;13,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;13,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.72799688513037, −12.13256641613510, −11.85237310228868, −11.51672668456082, −10.82973217460204, −10.54170403991867, −9.902322882888798, −9.780245065177993, −9.236295910519077, −8.511761901010927, −8.273161086951585, −7.934817158519330, −7.287712156017489, −6.886944162389289, −6.256104889778389, −5.966890958214367, −5.186420298737053, −5.000035795430123, −4.187787204194905, −3.791462155620443, −2.999270982137882, −2.367818454726312, −2.100460428428916, −1.308503365914371, −0.8729243002503930, 0, 0.8729243002503930, 1.308503365914371, 2.100460428428916, 2.367818454726312, 2.999270982137882, 3.791462155620443, 4.187787204194905, 5.000035795430123, 5.186420298737053, 5.966890958214367, 6.256104889778389, 6.886944162389289, 7.287712156017489, 7.934817158519330, 8.273161086951585, 8.511761901010927, 9.236295910519077, 9.780245065177993, 9.902322882888798, 10.54170403991867, 10.82973217460204, 11.51672668456082, 11.85237310228868, 12.13256641613510, 12.72799688513037

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