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 + 0.268·5-s + 6-s + 3.42·7-s + 8-s + 9-s + 0.268·10-s + 11-s + 12-s + 1.26·13-s + 3.42·14-s + 0.268·15-s + 16-s + 0.867·17-s + 18-s + 6.18·19-s + 0.268·20-s + 3.42·21-s + 22-s − 8.86·23-s + 24-s − 4.92·25-s + 1.26·26-s + 27-s + 3.42·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.120·5-s + 0.408·6-s + 1.29·7-s + 0.353·8-s + 0.333·9-s + 0.0850·10-s + 0.301·11-s + 0.288·12-s + 0.351·13-s + 0.915·14-s + 0.0694·15-s + 0.250·16-s + 0.210·17-s + 0.235·18-s + 1.41·19-s + 0.0601·20-s + 0.747·21-s + 0.213·22-s − 1.84·23-s + 0.204·24-s − 0.985·25-s + 0.248·26-s + 0.192·27-s + 0.647·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.896520278$
$L(\frac12)$  $\approx$  $4.896520278$
$L(\frac{3}{2})$   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 \( 1 - 0.268T + 5T^{2} \)
7 \( 1 - 3.42T + 7T^{2} \)
13 \( 1 - 1.26T + 13T^{2} \)
17 \( 1 - 0.867T + 17T^{2} \)
19 \( 1 - 6.18T + 19T^{2} \)
23 \( 1 + 8.86T + 23T^{2} \)
29 \( 1 - 1.57T + 29T^{2} \)
31 \( 1 + 6.59T + 31T^{2} \)
37 \( 1 - 8.55T + 37T^{2} \)
41 \( 1 - 3.44T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 - 5.99T + 47T^{2} \)
53 \( 1 - 4.13T + 53T^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
67 \( 1 + 0.380T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 2.36T + 73T^{2} \)
79 \( 1 + 8.37T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−8.158871890567133197650360066815, −7.75120761061341265800574477428, −7.16719083416261549912934665877, −5.77677909022724063222602645491, −5.72262413435157540712069016117, −4.36406794725729034724955942154, −4.09564601022005236552420561109, −2.98942932835518122744083189083, −2.04687185457531416279492577786, −1.26139052765123108472750141313, 1.26139052765123108472750141313, 2.04687185457531416279492577786, 2.98942932835518122744083189083, 4.09564601022005236552420561109, 4.36406794725729034724955942154, 5.72262413435157540712069016117, 5.77677909022724063222602645491, 7.16719083416261549912934665877, 7.75120761061341265800574477428, 8.158871890567133197650360066815

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