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Properties

Label 2-19890-1.1-c1-0-3
Degree $2$
Conductor $19890$
Sign $1$
Analytic cond. $158.822$
Root an. cond. $12.6024$
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 − 5-s − 4·7-s + 8-s − 10-s + 4·11-s − 13-s − 4·14-s + 16-s + 17-s − 8·19-s − 20-s + 4·22-s + 8·23-s + 25-s − 26-s − 4·28-s + 10·29-s − 8·31-s + 32-s + 34-s + 4·35-s − 6·37-s − 8·38-s − 40-s − 2·41-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 + 1.20·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.242·17-s − 1.83·19-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s − 0.196·26-s − 0.755·28-s + 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.171·34-s + 0.676·35-s − 0.986·37-s − 1.29·38-s − 0.158·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19890\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(158.822\)
Root analytic conductor: \(12.6024\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19890,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.239673640\)
\(L(\frac12)\) \(\approx\) \(2.239673640\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.46589184851520, −15.21986551415732, −14.48759540158344, −14.19513447728216, −13.30618146959567, −12.93195092861120, −12.32059195986241, −12.20541971407070, −11.25672750634377, −10.79008034318224, −10.22079334457315, −9.488996692514323, −8.958490657943021, −8.465700483368756, −7.507767967684223, −6.799427825751252, −6.555699625567637, −6.077136298989913, −5.052277899654717, −4.533709743040829, −3.711204464720889, −3.348051630365098, −2.636627734129269, −1.652894602464148, −0.5443975385278767, 0.5443975385278767, 1.652894602464148, 2.636627734129269, 3.348051630365098, 3.711204464720889, 4.533709743040829, 5.052277899654717, 6.077136298989913, 6.555699625567637, 6.799427825751252, 7.507767967684223, 8.465700483368756, 8.958490657943021, 9.488996692514323, 10.22079334457315, 10.79008034318224, 11.25672750634377, 12.20541971407070, 12.32059195986241, 12.93195092861120, 13.30618146959567, 14.19513447728216, 14.48759540158344, 15.21986551415732, 15.46589184851520

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