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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 3.63·5-s + 6-s − 1.65·7-s + 8-s + 9-s + 3.63·10-s + 11-s + 12-s + 6.43·13-s − 1.65·14-s + 3.63·15-s + 16-s − 1.11·17-s + 18-s − 5.55·19-s + 3.63·20-s − 1.65·21-s + 22-s − 3.64·23-s + 24-s + 8.22·25-s + 6.43·26-s + 27-s − 1.65·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.62·5-s + 0.408·6-s − 0.625·7-s + 0.353·8-s + 0.333·9-s + 1.14·10-s + 0.301·11-s + 0.288·12-s + 1.78·13-s − 0.442·14-s + 0.938·15-s + 0.250·16-s − 0.270·17-s + 0.235·18-s − 1.27·19-s + 0.813·20-s − 0.361·21-s + 0.213·22-s − 0.759·23-s + 0.204·24-s + 1.64·25-s + 1.26·26-s + 0.192·27-s − 0.312·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.276051889$
$L(\frac12)$  $\approx$  $5.276051889$
$L(\frac{3}{2})$   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 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 + 1.65T + 7T^{2} \)
13 \( 1 - 6.43T + 13T^{2} \)
17 \( 1 + 1.11T + 17T^{2} \)
19 \( 1 + 5.55T + 19T^{2} \)
23 \( 1 + 3.64T + 23T^{2} \)
29 \( 1 + 1.74T + 29T^{2} \)
31 \( 1 - 1.99T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + 3.65T + 41T^{2} \)
43 \( 1 - 6.64T + 43T^{2} \)
47 \( 1 - 11.2T + 47T^{2} \)
53 \( 1 + 7.64T + 53T^{2} \)
59 \( 1 + 0.0748T + 59T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 + 5.87T + 71T^{2} \)
73 \( 1 + 7.77T + 73T^{2} \)
79 \( 1 - 1.55T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 + 3.71T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.627871646178233920973516706498, −7.67118095269046701444332659029, −6.53436622417982006571654549277, −6.17082244401317710228942974282, −5.79322818005391816027610621739, −4.52375326167960590328848718817, −3.85813997286495880827037748348, −2.90249919544708770058440216347, −2.13745591278548257553611793246, −1.30295381433341018942114956649, 1.30295381433341018942114956649, 2.13745591278548257553611793246, 2.90249919544708770058440216347, 3.85813997286495880827037748348, 4.52375326167960590328848718817, 5.79322818005391816027610621739, 6.17082244401317710228942974282, 6.53436622417982006571654549277, 7.67118095269046701444332659029, 8.627871646178233920973516706498

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