↑

Properties

Label 2-6034-1.1-c1-0-176
Degree $2$
Conductor $6034$
Sign $1$
Analytic cond. $48.1817$
Root an. cond. $6.94130$
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 − 3·3-s + 4-s − 5-s + 3·6-s − 7-s − 8-s + 6·9-s + 10-s − 3·11-s − 3·12-s − 4·13-s + 14-s + 3·15-s + 16-s + 2·17-s − 6·18-s − 3·19-s − 20-s + 3·21-s + 3·22-s − 3·23-s + 3·24-s − 4·25-s + 4·26-s − 9·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 0.904·11-s − 0.866·12-s − 1.10·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 0.485·17-s − 1.41·18-s − 0.688·19-s − 0.223·20-s + 0.654·21-s + 0.639·22-s − 0.625·23-s + 0.612·24-s − 4/5·25-s + 0.784·26-s − 1.73·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6034\)    =    \(2 \cdot 7 \cdot 431\)
Sign: $1$
Analytic conductor: \(48.1817\)
Root analytic conductor: \(6.94130\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 6034,\ (\ :1/2),\ 1)\)

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 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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

−7.40984276921170137594542032795, −6.62203735660000235515565656805, −6.04331036787085984707660707268, −5.29579204346932426092329127781, −4.71442625075683181900320000783, −3.75372265152858173813383283329, −2.58545727434917695430675575620, −1.49867791679881499414611675912, 0, 0, 1.49867791679881499414611675912, 2.58545727434917695430675575620, 3.75372265152858173813383283329, 4.71442625075683181900320000783, 5.29579204346932426092329127781, 6.04331036787085984707660707268, 6.62203735660000235515565656805, 7.40984276921170137594542032795

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