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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s − 11-s + 13-s − 14-s + 16-s − 17-s − 7·19-s − 20-s + 22-s + 3·23-s + 25-s − 26-s + 28-s + 8·29-s − 5·31-s − 32-s + 34-s − 35-s + 5·37-s + 7·38-s + 40-s + 10·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.60·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s − 0.196·26-s + 0.188·28-s + 1.48·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.169·35-s + 0.821·37-s + 1.13·38-s + 0.158·40-s + 1.56·41-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  =  1
Selberg data  =  $(2,\ 16830,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

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

−16.16038522093195, −15.76862360539629, −15.07402256088265, −14.68970350306790, −14.13402319789308, −13.28215702919422, −12.69784724483732, −12.37515521334948, −11.40223963043811, −11.15236810435491, −10.59431586072650, −10.06367182694671, −9.223215775700059, −8.727982832654222, −8.281107682624322, −7.594560109032239, −7.156702813516730, −6.206132558672556, −5.993665477554194, −4.668096565152519, −4.528714814286349, −3.439439748535999, −2.704625120985872, −1.936800877464726, −1.008805138413670, 0, 1.008805138413670, 1.936800877464726, 2.704625120985872, 3.439439748535999, 4.528714814286349, 4.668096565152519, 5.993665477554194, 6.206132558672556, 7.156702813516730, 7.594560109032239, 8.281107682624322, 8.727982832654222, 9.223215775700059, 10.06367182694671, 10.59431586072650, 11.15236810435491, 11.40223963043811, 12.37515521334948, 12.69784724483732, 13.28215702919422, 14.13402319789308, 14.68970350306790, 15.07402256088265, 15.76862360539629, 16.16038522093195

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