Properties

Label 2-47096-1.1-c1-0-5
Degree $2$
Conductor $47096$
Sign $-1$
Analytic cond. $376.063$
Root an. cond. $19.3923$
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  − 2·3-s − 4·5-s + 7-s + 9-s + 8·15-s + 2·17-s + 2·19-s − 2·21-s + 8·23-s + 11·25-s + 4·27-s − 4·31-s − 4·35-s + 6·37-s + 2·41-s − 8·43-s − 4·45-s + 4·47-s + 49-s − 4·51-s − 10·53-s − 4·57-s + 6·59-s − 4·61-s + 63-s − 12·67-s − 16·69-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s + 0.485·17-s + 0.458·19-s − 0.436·21-s + 1.66·23-s + 11/5·25-s + 0.769·27-s − 0.718·31-s − 0.676·35-s + 0.986·37-s + 0.312·41-s − 1.21·43-s − 0.596·45-s + 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s − 0.512·61-s + 0.125·63-s − 1.46·67-s − 1.92·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(47096\)    =    \(2^{3} \cdot 7 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(376.063\)
Root analytic conductor: \(19.3923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{47096} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 47096,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
29 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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

−14.98797458164686, −14.55091210049263, −13.85793501162555, −13.04428160321893, −12.64894525589994, −11.98047140245545, −11.86073368229507, −11.19210638940063, −10.92384135803295, −10.57685435748113, −9.597290234321320, −9.079932352873968, −8.370733539041747, −7.912817998358269, −7.422119069301653, −6.865357616834648, −6.381606542615712, −5.469624574526234, −5.104312074230597, −4.572594131711036, −3.946766585497106, −3.266039507468151, −2.733525389556394, −1.371398022501222, −0.7727427258012689, 0, 0.7727427258012689, 1.371398022501222, 2.733525389556394, 3.266039507468151, 3.946766585497106, 4.572594131711036, 5.104312074230597, 5.469624574526234, 6.381606542615712, 6.865357616834648, 7.422119069301653, 7.912817998358269, 8.370733539041747, 9.079932352873968, 9.597290234321320, 10.57685435748113, 10.92384135803295, 11.19210638940063, 11.86073368229507, 11.98047140245545, 12.64894525589994, 13.04428160321893, 13.85793501162555, 14.55091210049263, 14.98797458164686

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