Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13 $
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 − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s + 13-s − 14-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s − 20-s + 21-s − 24-s + 25-s − 26-s + 27-s + 28-s + 6·29-s + 30-s + 8·31-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.267·14-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 + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2730\)    =    \(2 \cdot 3 \cdot 5 \cdot 7 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2730} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 2730,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.476679476$
$L(\frac12)$  $\approx$  $1.476679476$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;7,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
13 \( 1 - T \)
good11 \( 1 + p T^{2} \)
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 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−18.86358485032848, −18.24014871834496, −17.40535531062417, −17.26231841956669, −15.99830249224526, −15.77552874465422, −15.13798334330692, −14.40907869348274, −13.74430135220316, −12.98276101432399, −12.25237401092514, −11.53095274381319, −10.77018046591914, −10.37575346591360, −9.266815377283748, −8.811692992150160, −8.173682437008162, −7.553104713463467, −6.678869870378732, −6.051891941413320, −4.626383358130685, −4.137667519642438, −2.851561102567315, −2.138411992263997, −0.8188401717440957, 0.8188401717440957, 2.138411992263997, 2.851561102567315, 4.137667519642438, 4.626383358130685, 6.051891941413320, 6.678869870378732, 7.553104713463467, 8.173682437008162, 8.811692992150160, 9.266815377283748, 10.37575346591360, 10.77018046591914, 11.53095274381319, 12.25237401092514, 12.98276101432399, 13.74430135220316, 14.40907869348274, 15.13798334330692, 15.77552874465422, 15.99830249224526, 17.26231841956669, 17.40535531062417, 18.24014871834496, 18.86358485032848

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