Properties

Degree $2$
Conductor $14994$
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 − 8-s − 6·11-s − 2·13-s + 16-s − 17-s + 4·19-s + 6·22-s − 5·25-s + 2·26-s + 4·31-s − 32-s + 34-s − 4·37-s − 4·38-s + 6·41-s + 8·43-s − 6·44-s + 5·50-s − 2·52-s + 6·53-s + 4·61-s − 4·62-s + 64-s + 8·67-s − 68-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 1.27·22-s − 25-s + 0.392·26-s + 0.718·31-s − 0.176·32-s + 0.171·34-s − 0.657·37-s − 0.648·38-s + 0.937·41-s + 1.21·43-s − 0.904·44-s + 0.707·50-s − 0.277·52-s + 0.824·53-s + 0.512·61-s − 0.508·62-s + 1/8·64-s + 0.977·67-s − 0.121·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14994 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14994\)    =    \(2 \cdot 3^{2} \cdot 7^{2} \cdot 17\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{14994} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14994,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
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 \( 1 + 14 T + p T^{2} \)
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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

−16.19735639155593, −15.81894486354278, −15.47614175415650, −14.84512222482307, −13.96452409245311, −13.68270221440269, −12.80120177746566, −12.50789324029013, −11.65460829521950, −11.25789418312381, −10.44541909239074, −10.15916824343610, −9.545798525295212, −8.907728772945927, −8.160618185897864, −7.663088571939708, −7.318513675843698, −6.456107145824030, −5.623387660943705, −5.240033842136342, −4.391358871370364, −3.443174138102097, −2.580109971514423, −2.200364964798855, −0.9422387846645484, 0, 0.9422387846645484, 2.200364964798855, 2.580109971514423, 3.443174138102097, 4.391358871370364, 5.240033842136342, 5.623387660943705, 6.456107145824030, 7.318513675843698, 7.663088571939708, 8.160618185897864, 8.907728772945927, 9.545798525295212, 10.15916824343610, 10.44541909239074, 11.25789418312381, 11.65460829521950, 12.50789324029013, 12.80120177746566, 13.68270221440269, 13.96452409245311, 14.84512222482307, 15.47614175415650, 15.81894486354278, 16.19735639155593

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