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 + 2.39·3-s + 4-s + 1.79·5-s − 2.39·6-s − 7-s − 8-s + 2.74·9-s − 1.79·10-s + 4.02·11-s + 2.39·12-s − 4.04·13-s + 14-s + 4.29·15-s + 16-s − 2.85·17-s − 2.74·18-s + 7.25·19-s + 1.79·20-s − 2.39·21-s − 4.02·22-s − 3.34·23-s − 2.39·24-s − 1.78·25-s + 4.04·26-s − 0.610·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.38·3-s + 0.5·4-s + 0.801·5-s − 0.978·6-s − 0.377·7-s − 0.353·8-s + 0.915·9-s − 0.566·10-s + 1.21·11-s + 0.691·12-s − 1.12·13-s + 0.267·14-s + 1.10·15-s + 0.250·16-s − 0.691·17-s − 0.647·18-s + 1.66·19-s + 0.400·20-s − 0.523·21-s − 0.858·22-s − 0.696·23-s − 0.489·24-s − 0.357·25-s + 0.792·26-s − 0.117·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.896743008$
$L(\frac12)$  $\approx$  $2.896743008$
$L(\frac{3}{2})$   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 \( 1 - 2.39T + 3T^{2} \)
5 \( 1 - 1.79T + 5T^{2} \)
11 \( 1 - 4.02T + 11T^{2} \)
13 \( 1 + 4.04T + 13T^{2} \)
17 \( 1 + 2.85T + 17T^{2} \)
19 \( 1 - 7.25T + 19T^{2} \)
23 \( 1 + 3.34T + 23T^{2} \)
29 \( 1 - 9.46T + 29T^{2} \)
31 \( 1 - 5.02T + 31T^{2} \)
37 \( 1 - 2.96T + 37T^{2} \)
41 \( 1 + 0.0546T + 41T^{2} \)
43 \( 1 - 2.62T + 43T^{2} \)
47 \( 1 + 5.09T + 47T^{2} \)
53 \( 1 - 6.76T + 53T^{2} \)
59 \( 1 - 2.56T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 + 7.03T + 67T^{2} \)
71 \( 1 - 11.2T + 71T^{2} \)
73 \( 1 - 16.6T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 - 3.21T + 83T^{2} \)
89 \( 1 + 3.96T + 89T^{2} \)
97 \( 1 - 1.99T + 97T^{2} \)
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

−8.092677629595381440005658244711, −7.62546301602840147235309847031, −6.72974226871416845954210574512, −6.29750986974879993511201132921, −5.23524565699603642589104544205, −4.24365721879309426492179818891, −3.33651468774425929570545223533, −2.60269640555005906100643718085, −2.00828766628705601697018006516, −0.951335473960372593942065499980, 0.951335473960372593942065499980, 2.00828766628705601697018006516, 2.60269640555005906100643718085, 3.33651468774425929570545223533, 4.24365721879309426492179818891, 5.23524565699603642589104544205, 6.29750986974879993511201132921, 6.72974226871416845954210574512, 7.62546301602840147235309847031, 8.092677629595381440005658244711

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