Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 11 \cdot 61 $
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 + 3-s + 4-s + 2.48·5-s + 6-s + 3.05·7-s + 8-s + 9-s + 2.48·10-s + 11-s + 12-s − 0.563·13-s + 3.05·14-s + 2.48·15-s + 16-s − 7.92·17-s + 18-s + 2.52·19-s + 2.48·20-s + 3.05·21-s + 22-s + 5.23·23-s + 24-s + 1.15·25-s − 0.563·26-s + 27-s + 3.05·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.10·5-s + 0.408·6-s + 1.15·7-s + 0.353·8-s + 0.333·9-s + 0.784·10-s + 0.301·11-s + 0.288·12-s − 0.156·13-s + 0.816·14-s + 0.640·15-s + 0.250·16-s − 1.92·17-s + 0.235·18-s + 0.578·19-s + 0.554·20-s + 0.666·21-s + 0.213·22-s + 1.09·23-s + 0.204·24-s + 0.230·25-s − 0.110·26-s + 0.192·27-s + 0.577·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4026} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.447199663$
$L(\frac12)$  $\approx$  $5.447199663$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;61\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;61\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
61 \( 1 + T \)
good5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 3.05T + 7T^{2} \)
13 \( 1 + 0.563T + 13T^{2} \)
17 \( 1 + 7.92T + 17T^{2} \)
19 \( 1 - 2.52T + 19T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 + 6.06T + 29T^{2} \)
31 \( 1 - 9.71T + 31T^{2} \)
37 \( 1 + 0.209T + 37T^{2} \)
41 \( 1 + 8.94T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
67 \( 1 - 0.884T + 67T^{2} \)
71 \( 1 + 0.510T + 71T^{2} \)
73 \( 1 + 9.15T + 73T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 - 9.91T + 83T^{2} \)
89 \( 1 + 6.04T + 89T^{2} \)
97 \( 1 + 1.64T + 97T^{2} \)
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

−8.558589449845641110960928792118, −7.61848087654334840687721836918, −6.87960458586713905159273916600, −6.22567101423073458569290779430, −5.27794810469212968583883142871, −4.74182405370605766205846197207, −3.97581725997408098800573050162, −2.78390027195843155630066161382, −2.12162360611696354053190322571, −1.35201793970015482301976063819, 1.35201793970015482301976063819, 2.12162360611696354053190322571, 2.78390027195843155630066161382, 3.97581725997408098800573050162, 4.74182405370605766205846197207, 5.27794810469212968583883142871, 6.22567101423073458569290779430, 6.87960458586713905159273916600, 7.61848087654334840687721836918, 8.558589449845641110960928792118

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