Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17 $
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 + 4-s − 5-s − 4·7-s + 8-s − 10-s − 11-s + 2·13-s − 4·14-s + 16-s + 17-s − 4·19-s − 20-s − 22-s + 25-s + 2·26-s − 4·28-s + 6·29-s + 8·31-s + 32-s + 34-s + 4·35-s − 10·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.554·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s − 0.213·22-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.676·35-s − 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(16830\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 11 \cdot 17\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{16830} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.853533468$
$L(\frac12)$  $\approx$  $1.853533468$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−15.82296427401338, −15.46717689423417, −14.89744351512924, −14.08531187468509, −13.63015460362844, −13.15258269007756, −12.56458814877471, −12.17180605060257, −11.63204330693858, −10.77706940296985, −10.38148146676193, −9.812456332050219, −9.125362882931759, −8.252720698398286, −8.021081960164316, −6.828819674999402, −6.581572993294051, −6.183394755534541, −5.157857433621647, −4.692198569320017, −3.732075949425965, −3.340487610785958, −2.738774837531712, −1.727219870004097, −0.5021688061542489, 0.5021688061542489, 1.727219870004097, 2.738774837531712, 3.340487610785958, 3.732075949425965, 4.692198569320017, 5.157857433621647, 6.183394755534541, 6.581572993294051, 6.828819674999402, 8.021081960164316, 8.252720698398286, 9.125362882931759, 9.812456332050219, 10.38148146676193, 10.77706940296985, 11.63204330693858, 12.17180605060257, 12.56458814877471, 13.15258269007756, 13.63015460362844, 14.08531187468509, 14.89744351512924, 15.46717689423417, 15.82296427401338

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