Properties

Degree $2$
Conductor $84966$
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 − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 6·13-s + 2·15-s + 16-s + 18-s + 4·19-s − 2·20-s + 4·22-s − 8·23-s − 24-s − 25-s − 6·26-s − 27-s + 2·29-s + 2·30-s + 32-s − 4·33-s + 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.516·15-s + 1/4·16-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s − 1.66·23-s − 0.204·24-s − 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s + 0.176·32-s − 0.696·33-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84966\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{84966} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 84966,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
17 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.27640114607469, −13.69790765079136, −13.20066206236938, −12.39220018137305, −12.12998307769760, −11.82068782940482, −11.51810032687050, −10.92445318935126, −10.13604535478320, −9.773955567109087, −9.411119735119491, −8.467877472855487, −7.924513137605369, −7.489255661210195, −6.995364485348405, −6.457198061579472, −5.944641925767685, −5.258321352047645, −4.814874282641343, −4.095626606948838, −3.937364724229683, −3.135875733391663, −2.403401334294822, −1.717822133723067, −0.8321233335871864, 0, 0.8321233335871864, 1.717822133723067, 2.403401334294822, 3.135875733391663, 3.937364724229683, 4.095626606948838, 4.814874282641343, 5.258321352047645, 5.944641925767685, 6.457198061579472, 6.995364485348405, 7.489255661210195, 7.924513137605369, 8.467877472855487, 9.411119735119491, 9.773955567109087, 10.13604535478320, 10.92445318935126, 11.51810032687050, 11.82068782940482, 12.12998307769760, 12.39220018137305, 13.20066206236938, 13.69790765079136, 14.27640114607469

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