Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 431 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2.83·3-s + 4-s + 0.330·5-s + 2.83·6-s − 7-s − 8-s + 5.05·9-s − 0.330·10-s + 3.07·11-s − 2.83·12-s + 2.69·13-s + 14-s − 0.936·15-s + 16-s − 6.41·17-s − 5.05·18-s + 3.15·19-s + 0.330·20-s + 2.83·21-s − 3.07·22-s − 6.50·23-s + 2.83·24-s − 4.89·25-s − 2.69·26-s − 5.84·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.63·3-s + 0.5·4-s + 0.147·5-s + 1.15·6-s − 0.377·7-s − 0.353·8-s + 1.68·9-s − 0.104·10-s + 0.928·11-s − 0.819·12-s + 0.746·13-s + 0.267·14-s − 0.241·15-s + 0.250·16-s − 1.55·17-s − 1.19·18-s + 0.723·19-s + 0.0737·20-s + 0.619·21-s − 0.656·22-s − 1.35·23-s + 0.579·24-s − 0.978·25-s − 0.528·26-s − 1.12·27-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

\( d \)  =  \(2\)
\( N \)  =  \(6034\)    =    \(2 \cdot 7 \cdot 431\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6325082220$
$L(\frac12)$  $\approx$  $0.6325082220$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;431\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;431\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 + T \)
431 \( 1 - T \)
good3 \( 1 + 2.83T + 3T^{2} \)
5 \( 1 - 0.330T + 5T^{2} \)
11 \( 1 - 3.07T + 11T^{2} \)
13 \( 1 - 2.69T + 13T^{2} \)
17 \( 1 + 6.41T + 17T^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 + 6.50T + 23T^{2} \)
29 \( 1 - 3.58T + 29T^{2} \)
31 \( 1 + 0.453T + 31T^{2} \)
37 \( 1 - 0.0743T + 37T^{2} \)
41 \( 1 - 9.21T + 41T^{2} \)
43 \( 1 - 7.47T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 - 3.91T + 53T^{2} \)
59 \( 1 - 7.17T + 59T^{2} \)
61 \( 1 + 7.56T + 61T^{2} \)
67 \( 1 - 1.60T + 67T^{2} \)
71 \( 1 + 5.69T + 71T^{2} \)
73 \( 1 + 6.01T + 73T^{2} \)
79 \( 1 + 5.72T + 79T^{2} \)
83 \( 1 - 9.56T + 83T^{2} \)
89 \( 1 + 8.49T + 89T^{2} \)
97 \( 1 - 1.71T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.015050801805333997999219662615, −7.14565036408826449910405058946, −6.60175405600989564573684461691, −5.93918567888322093237869955226, −5.67614720590483500766038016041, −4.34414097941987469208730280141, −3.97115994220480043279460330172, −2.50023319411368462537445397048, −1.44252676124240884623035271505, −0.54185854538306070155732138915, 0.54185854538306070155732138915, 1.44252676124240884623035271505, 2.50023319411368462537445397048, 3.97115994220480043279460330172, 4.34414097941987469208730280141, 5.67614720590483500766038016041, 5.93918567888322093237869955226, 6.60175405600989564573684461691, 7.14565036408826449910405058946, 8.015050801805333997999219662615

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