Properties

Label 2-1850-1.1-c1-0-52
Degree $2$
Conductor $1850$
Sign $-1$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 3·9-s − 4·11-s − 2·13-s + 16-s + 2·17-s − 3·18-s − 4·19-s − 4·22-s − 2·26-s − 6·29-s − 4·31-s + 32-s + 2·34-s − 3·36-s + 37-s − 4·38-s − 6·41-s − 4·43-s − 4·44-s + 8·47-s − 7·49-s − 2·52-s − 10·53-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 1.20·11-s − 0.554·13-s + 1/4·16-s + 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.852·22-s − 0.392·26-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.342·34-s − 1/2·36-s + 0.164·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.16·47-s − 49-s − 0.277·52-s − 1.37·53-s − 0.787·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $-1$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
37 \( 1 - T \)
good3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.712726996024504289860030578877, −7.985100866972308578412355215355, −7.26997651195770461478664014699, −6.27355954369629806358703833439, −5.46325332967485333134031287798, −4.95477554784516684995777664760, −3.76639174593054485317551916841, −2.87123189491779009496507990789, −2.02250478001830636560137396847, 0, 2.02250478001830636560137396847, 2.87123189491779009496507990789, 3.76639174593054485317551916841, 4.95477554784516684995777664760, 5.46325332967485333134031287798, 6.27355954369629806358703833439, 7.26997651195770461478664014699, 7.985100866972308578412355215355, 8.712726996024504289860030578877

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