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 − 2.58·3-s + 4-s − 2.90·5-s + 2.58·6-s − 7-s − 8-s + 3.65·9-s + 2.90·10-s + 1.98·11-s − 2.58·12-s − 1.13·13-s + 14-s + 7.49·15-s + 16-s + 6.51·17-s − 3.65·18-s + 6.40·19-s − 2.90·20-s + 2.58·21-s − 1.98·22-s + 2.52·23-s + 2.58·24-s + 3.42·25-s + 1.13·26-s − 1.70·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.48·3-s + 0.5·4-s − 1.29·5-s + 1.05·6-s − 0.377·7-s − 0.353·8-s + 1.21·9-s + 0.917·10-s + 0.597·11-s − 0.744·12-s − 0.315·13-s + 0.267·14-s + 1.93·15-s + 0.250·16-s + 1.58·17-s − 0.862·18-s + 1.46·19-s − 0.649·20-s + 0.563·21-s − 0.422·22-s + 0.527·23-s + 0.526·24-s + 0.685·25-s + 0.222·26-s − 0.327·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6538590473$
$L(\frac12)$  $\approx$  $0.6538590473$
$L(\frac{3}{2})$   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 \( 1 + 2.58T + 3T^{2} \)
5 \( 1 + 2.90T + 5T^{2} \)
11 \( 1 - 1.98T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 - 6.40T + 19T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 - 2.57T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 - 2.00T + 37T^{2} \)
41 \( 1 - 2.58T + 41T^{2} \)
43 \( 1 + 0.614T + 43T^{2} \)
47 \( 1 - 6.59T + 47T^{2} \)
53 \( 1 - 4.77T + 53T^{2} \)
59 \( 1 + 4.80T + 59T^{2} \)
61 \( 1 - 5.92T + 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 - 7.79T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 4.85T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 0.899T + 89T^{2} \)
97 \( 1 - 7.62T + 97T^{2} \)
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

−7.83644439361281844002775753946, −7.39561016506271007300760164368, −6.78895258954011237492512465846, −5.98576462997948841931469835114, −5.35617852426957797980580249134, −4.54797504338195638238710801433, −3.64157741286351606633163316408, −2.89484327729758742916168707671, −1.13959688450968355724241334381, −0.64015562112288881359343420960, 0.64015562112288881359343420960, 1.13959688450968355724241334381, 2.89484327729758742916168707671, 3.64157741286351606633163316408, 4.54797504338195638238710801433, 5.35617852426957797980580249134, 5.98576462997948841931469835114, 6.78895258954011237492512465846, 7.39561016506271007300760164368, 7.83644439361281844002775753946

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