Properties

Label 2-14994-1.1-c1-0-11
Degree $2$
Conductor $14994$
Sign $1$
Analytic cond. $119.727$
Root an. cond. $10.9420$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 3·5-s − 8-s − 3·10-s − 2·11-s + 16-s − 17-s − 3·19-s + 3·20-s + 2·22-s + 5·23-s + 4·25-s − 6·29-s − 32-s + 34-s + 3·37-s + 3·38-s − 3·40-s − 43-s − 2·44-s − 5·46-s + 6·47-s − 4·50-s + 6·53-s − 6·55-s + 6·58-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s − 0.603·11-s + 1/4·16-s − 0.242·17-s − 0.688·19-s + 0.670·20-s + 0.426·22-s + 1.04·23-s + 4/5·25-s − 1.11·29-s − 0.176·32-s + 0.171·34-s + 0.493·37-s + 0.486·38-s − 0.474·40-s − 0.152·43-s − 0.301·44-s − 0.737·46-s + 0.875·47-s − 0.565·50-s + 0.824·53-s − 0.809·55-s + 0.787·58-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: $1$
Analytic conductor: \(119.727\)
Root analytic conductor: \(10.9420\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14994} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 14994,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.798771552\)
\(L(\frac12)\) \(\approx\) \(1.798771552\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 5 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−16.30442670454592, −15.41605956897958, −15.06243412836935, −14.45695512560977, −13.75100783806929, −13.15384796890490, −12.93106164346666, −12.09793777212632, −11.37255721505961, −10.67784051032188, −10.46076081847915, −9.691004864481392, −9.222973552238763, −8.775117318637834, −8.012818170679303, −7.339761299742857, −6.717483178229709, −6.068322272505175, −5.526136703191188, −4.902167745602219, −3.927742905513748, −2.920918300296958, −2.282736226898327, −1.686469652818581, −0.6449703231318714, 0.6449703231318714, 1.686469652818581, 2.282736226898327, 2.920918300296958, 3.927742905513748, 4.902167745602219, 5.526136703191188, 6.068322272505175, 6.717483178229709, 7.339761299742857, 8.012818170679303, 8.775117318637834, 9.222973552238763, 9.691004864481392, 10.46076081847915, 10.67784051032188, 11.37255721505961, 12.09793777212632, 12.93106164346666, 13.15384796890490, 13.75100783806929, 14.45695512560977, 15.06243412836935, 15.41605956897958, 16.30442670454592

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