Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \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 − 4·7-s + 8-s + 9-s + 10-s − 11-s + 12-s + 13-s − 4·14-s + 15-s + 16-s − 6·17-s + 18-s − 4·19-s + 20-s − 4·21-s − 22-s + 24-s + 25-s + 26-s + 27-s − 4·28-s − 6·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-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 − 1.06·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4290\)    =    \(2 \cdot 3 \cdot 5 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4290} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4290,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−18.36242226746781, −17.76546086623850, −16.71547894319516, −16.52581455374119, −15.61287759926752, −15.28516835027623, −14.65463931010428, −13.76604751970961, −13.28528955027167, −12.97256942979779, −12.44290122098784, −11.42285808001958, −10.72797145021202, −10.12641977539352, −9.334593252105868, −8.902114190924558, −7.987088704690405, −6.995240408841838, −6.546305485938607, −5.932768691314816, −4.991193066799859, −4.047675459441242, −3.420717906982489, −2.574752629618946, −1.846662494171753, 0, 1.846662494171753, 2.574752629618946, 3.420717906982489, 4.047675459441242, 4.991193066799859, 5.932768691314816, 6.546305485938607, 6.995240408841838, 7.987088704690405, 8.902114190924558, 9.334593252105868, 10.12641977539352, 10.72797145021202, 11.42285808001958, 12.44290122098784, 12.97256942979779, 13.28528955027167, 13.76604751970961, 14.65463931010428, 15.28516835027623, 15.61287759926752, 16.52581455374119, 16.71547894319516, 17.76546086623850, 18.36242226746781

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