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 − 2.88·5-s + 6-s − 2.96·7-s + 8-s + 9-s − 2.88·10-s + 11-s + 12-s + 1.99·13-s − 2.96·14-s − 2.88·15-s + 16-s + 3.32·17-s + 18-s − 5.75·19-s − 2.88·20-s − 2.96·21-s + 22-s + 7.24·23-s + 24-s + 3.32·25-s + 1.99·26-s + 27-s − 2.96·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.29·5-s + 0.408·6-s − 1.12·7-s + 0.353·8-s + 0.333·9-s − 0.912·10-s + 0.301·11-s + 0.288·12-s + 0.553·13-s − 0.792·14-s − 0.744·15-s + 0.250·16-s + 0.807·17-s + 0.235·18-s − 1.32·19-s − 0.645·20-s − 0.647·21-s + 0.213·22-s + 1.51·23-s + 0.204·24-s + 0.665·25-s + 0.391·26-s + 0.192·27-s − 0.560·28-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  =  $(2,\ 4026,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.576083437$
$L(\frac12)$  $\approx$  $2.576083437$
$L(\frac{3}{2})$   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 \( 1 + 2.88T + 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
13 \( 1 - 1.99T + 13T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 + 5.75T + 19T^{2} \)
23 \( 1 - 7.24T + 23T^{2} \)
29 \( 1 + 8.65T + 29T^{2} \)
31 \( 1 - 6.36T + 31T^{2} \)
37 \( 1 - 0.888T + 37T^{2} \)
41 \( 1 + 8.21T + 41T^{2} \)
43 \( 1 - 8.75T + 43T^{2} \)
47 \( 1 - 7.18T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 - 4.60T + 59T^{2} \)
67 \( 1 + 0.148T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 - 0.842T + 79T^{2} \)
83 \( 1 + 2.03T + 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 - 15.7T + 97T^{2} \)
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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

−8.399970640378447314648314399544, −7.60005032790609629931850675139, −6.97434770558838652099050607941, −6.34929182930948687824215005634, −5.42222603293876137837773705075, −4.35488223116745538461825571228, −3.72222142406546332745283236241, −3.30658820668560213808764560751, −2.32044253992205971673530168289, −0.795915076819202164711666993086, 0.795915076819202164711666993086, 2.32044253992205971673530168289, 3.30658820668560213808764560751, 3.72222142406546332745283236241, 4.35488223116745538461825571228, 5.42222603293876137837773705075, 6.34929182930948687824215005634, 6.97434770558838652099050607941, 7.60005032790609629931850675139, 8.399970640378447314648314399544

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