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 − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 3·13-s + 2·14-s + 15-s + 16-s + 8·17-s + 18-s + 4·19-s − 20-s − 2·21-s − 22-s + 4·23-s − 24-s − 4·25-s − 3·26-s − 27-s + 2·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.832·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s − 4/5·25-s − 0.588·26-s − 0.192·27-s + 0.377·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.540088498$
$L(\frac12)$  $\approx$  $2.540088498$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 + T \)
61 \( 1 + T \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 17 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−8.030512474930672716407714936515, −7.56946666189895832614022757751, −7.16705937386315268885077779613, −5.86956618399844379849712956796, −5.43460548429938109957373694760, −4.84023254455917646844177651898, −3.91145483661513441251018140705, −3.16184738395861988681461822456, −2.00694054467810328720814652990, −0.868596189690227387087423691586, 0.868596189690227387087423691586, 2.00694054467810328720814652990, 3.16184738395861988681461822456, 3.91145483661513441251018140705, 4.84023254455917646844177651898, 5.43460548429938109957373694760, 5.86956618399844379849712956796, 7.16705937386315268885077779613, 7.56946666189895832614022757751, 8.030512474930672716407714936515

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