Properties

Label 2-7942-1.1-c1-0-283
Degree $2$
Conductor $7942$
Sign $1$
Analytic cond. $63.4171$
Root an. cond. $7.96349$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 2·5-s − 6-s − 3·7-s − 8-s − 2·9-s + 2·10-s − 11-s + 12-s − 13-s + 3·14-s − 2·15-s + 16-s − 7·17-s + 2·18-s − 2·20-s − 3·21-s + 22-s − 5·23-s − 24-s − 25-s + 26-s − 5·27-s − 3·28-s − 29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.801·14-s − 0.516·15-s + 1/4·16-s − 1.69·17-s + 0.471·18-s − 0.447·20-s − 0.654·21-s + 0.213·22-s − 1.04·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.962·27-s − 0.566·28-s − 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7942\)    =    \(2 \cdot 11 \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(63.4171\)
Root analytic conductor: \(7.96349\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 7942,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
11 \( 1 + T \)
19 \( 1 \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
23 \( 1 + 5 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−7.47056361801104528213433887185, −6.51664925203855857429775517231, −6.07767886450910058758532668441, −5.03256507095952604569952374687, −3.96437100151690653774741793898, −3.46193594100619719426638672644, −2.61804273768923765457352807756, −1.92629873695338160262052084852, 0, 0, 1.92629873695338160262052084852, 2.61804273768923765457352807756, 3.46193594100619719426638672644, 3.96437100151690653774741793898, 5.03256507095952604569952374687, 6.07767886450910058758532668441, 6.51664925203855857429775517231, 7.47056361801104528213433887185

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