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 + 2.10·3-s + 4-s + 4.16·5-s − 2.10·6-s − 7-s − 8-s + 1.41·9-s − 4.16·10-s − 3.72·11-s + 2.10·12-s − 3.37·13-s + 14-s + 8.75·15-s + 16-s + 1.77·17-s − 1.41·18-s + 0.724·19-s + 4.16·20-s − 2.10·21-s + 3.72·22-s + 1.41·23-s − 2.10·24-s + 12.3·25-s + 3.37·26-s − 3.33·27-s − 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.21·3-s + 0.5·4-s + 1.86·5-s − 0.857·6-s − 0.377·7-s − 0.353·8-s + 0.471·9-s − 1.31·10-s − 1.12·11-s + 0.606·12-s − 0.934·13-s + 0.267·14-s + 2.26·15-s + 0.250·16-s + 0.430·17-s − 0.333·18-s + 0.166·19-s + 0.931·20-s − 0.458·21-s + 0.795·22-s + 0.294·23-s − 0.428·24-s + 2.47·25-s + 0.661·26-s − 0.641·27-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  =  $(2,\ 6034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.926120539$
$L(\frac12)$  $\approx$  $2.926120539$
$L(\frac{3}{2})$   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 \( 1 - 2.10T + 3T^{2} \)
5 \( 1 - 4.16T + 5T^{2} \)
11 \( 1 + 3.72T + 11T^{2} \)
13 \( 1 + 3.37T + 13T^{2} \)
17 \( 1 - 1.77T + 17T^{2} \)
19 \( 1 - 0.724T + 19T^{2} \)
23 \( 1 - 1.41T + 23T^{2} \)
29 \( 1 + 0.0884T + 29T^{2} \)
31 \( 1 - 6.94T + 31T^{2} \)
37 \( 1 - 3.16T + 37T^{2} \)
41 \( 1 - 8.67T + 41T^{2} \)
43 \( 1 - 5.90T + 43T^{2} \)
47 \( 1 - 9.49T + 47T^{2} \)
53 \( 1 + 0.0494T + 53T^{2} \)
59 \( 1 - 9.33T + 59T^{2} \)
61 \( 1 - 1.98T + 61T^{2} \)
67 \( 1 - 4.62T + 67T^{2} \)
71 \( 1 - 2.75T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 + 4.76T + 79T^{2} \)
83 \( 1 + 5.90T + 83T^{2} \)
89 \( 1 + 0.395T + 89T^{2} \)
97 \( 1 - 2.82T + 97T^{2} \)
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

−8.227607768501390005884183378772, −7.47773846080084615947007280951, −6.89266747366746721896157330316, −5.83300331863368148764939989831, −5.54641152638462950985556331353, −4.44625275851105727920911241196, −3.04369505760300258845432342925, −2.56261419375124475929545890678, −2.17224628559125214106977477244, −0.938636392979836618123654551160, 0.938636392979836618123654551160, 2.17224628559125214106977477244, 2.56261419375124475929545890678, 3.04369505760300258845432342925, 4.44625275851105727920911241196, 5.54641152638462950985556331353, 5.83300331863368148764939989831, 6.89266747366746721896157330316, 7.47773846080084615947007280951, 8.227607768501390005884183378772

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