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 − 4·7-s − 8-s + 10-s − 11-s − 2·13-s + 4·14-s + 16-s + 17-s − 2·19-s − 20-s + 22-s + 6·23-s + 25-s + 2·26-s − 4·28-s + 2·29-s − 8·31-s − 32-s − 34-s + 4·35-s − 6·37-s + 2·38-s + 40-s − 4·41-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 − 0.458·19-s − 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.392·26-s − 0.755·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.676·35-s − 0.986·37-s + 0.324·38-s + 0.158·40-s − 0.624·41-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 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.28177390822017, −15.72461954520903, −15.21373296230019, −14.72644747923442, −14.00079030832883, −13.18559520489504, −12.75186797886010, −12.39581413059338, −11.64818924523338, −11.06259621958392, −10.34871082074274, −10.06105573484414, −9.311692723617514, −8.875526458195747, −8.312899428178868, −7.337680858493628, −7.121494321552921, −6.524298567380775, −5.707536851250094, −5.127797763076898, −4.101980832981608, −3.397648531519361, −2.847608927732542, −2.041676296840032, −0.8029626969960879, 0, 0.8029626969960879, 2.041676296840032, 2.847608927732542, 3.397648531519361, 4.101980832981608, 5.127797763076898, 5.707536851250094, 6.524298567380775, 7.121494321552921, 7.337680858493628, 8.312899428178868, 8.875526458195747, 9.311692723617514, 10.06105573484414, 10.34871082074274, 11.06259621958392, 11.64818924523338, 12.39581413059338, 12.75186797886010, 13.18559520489504, 14.00079030832883, 14.72644747923442, 15.21373296230019, 15.72461954520903, 16.28177390822017

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