Properties

Label 2-19110-1.1-c1-0-34
Degree $2$
Conductor $19110$
Sign $1$
Analytic cond. $152.594$
Root an. cond. $12.3528$
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 + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s + 15-s + 16-s − 8·17-s + 18-s + 6·19-s + 20-s + 4·22-s + 6·23-s + 24-s + 25-s + 26-s + 27-s − 4·29-s + 30-s + 32-s + 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.742·29-s + 0.182·30-s + 0.176·32-s + 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(19110\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(152.594\)
Root analytic conductor: \(12.3528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{19110} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 19110,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.988374799\)
\(L(\frac12)\) \(\approx\) \(5.988374799\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−15.57639878654153, −15.17698284338397, −14.38065651627060, −14.19600715666065, −13.55014039482340, −13.06044700000217, −12.71380395382446, −11.82345006996817, −11.26588089614132, −11.03769840286876, −10.04637918130839, −9.515262137996659, −8.904818604889350, −8.627117613402736, −7.523584162589416, −7.108524516833185, −6.443463975707606, −6.030745216297070, −4.944404430358017, −4.728114292909355, −3.662318853926559, −3.391981349142410, −2.392943928335864, −1.793298926657252, −0.9163496015979278, 0.9163496015979278, 1.793298926657252, 2.392943928335864, 3.391981349142410, 3.662318853926559, 4.728114292909355, 4.944404430358017, 6.030745216297070, 6.443463975707606, 7.108524516833185, 7.523584162589416, 8.627117613402736, 8.904818604889350, 9.515262137996659, 10.04637918130839, 11.03769840286876, 11.26588089614132, 11.82345006996817, 12.71380395382446, 13.06044700000217, 13.55014039482340, 14.19600715666065, 14.38065651627060, 15.17698284338397, 15.57639878654153

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