Properties

Label 2-4026-1.1-c1-0-65
Degree $2$
Conductor $4026$
Sign $-1$
Analytic cond. $32.1477$
Root an. cond. $5.66990$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 1.66·5-s − 6-s − 2.43·7-s − 8-s + 9-s + 1.66·10-s − 11-s + 12-s + 6.41·13-s + 2.43·14-s − 1.66·15-s + 16-s − 0.265·17-s − 18-s − 4.82·19-s − 1.66·20-s − 2.43·21-s + 22-s + 5.37·23-s − 24-s − 2.22·25-s − 6.41·26-s + 27-s − 2.43·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.744·5-s − 0.408·6-s − 0.920·7-s − 0.353·8-s + 0.333·9-s + 0.526·10-s − 0.301·11-s + 0.288·12-s + 1.78·13-s + 0.650·14-s − 0.429·15-s + 0.250·16-s − 0.0644·17-s − 0.235·18-s − 1.10·19-s − 0.372·20-s − 0.531·21-s + 0.213·22-s + 1.12·23-s − 0.204·24-s − 0.445·25-s − 1.25·26-s + 0.192·27-s − 0.460·28-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

Degree: \(2\)
Conductor: \(4026\)    =    \(2 \cdot 3 \cdot 11 \cdot 61\)
Sign: $-1$
Analytic conductor: \(32.1477\)
Root analytic conductor: \(5.66990\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4026,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
11 \( 1 + T \)
61 \( 1 - T \)
good5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
13 \( 1 - 6.41T + 13T^{2} \)
17 \( 1 + 0.265T + 17T^{2} \)
19 \( 1 + 4.82T + 19T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 2.75T + 29T^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 + 6.27T + 37T^{2} \)
41 \( 1 - 7.91T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 - 0.474T + 47T^{2} \)
53 \( 1 + 10.2T + 53T^{2} \)
59 \( 1 - 10.8T + 59T^{2} \)
67 \( 1 - 1.15T + 67T^{2} \)
71 \( 1 - 6.34T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 - 7.07T + 83T^{2} \)
89 \( 1 - 4.10T + 89T^{2} \)
97 \( 1 - 4.25T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.315583506182269107091838909957, −7.50371880921043911309068608823, −6.65095096406869249314131224096, −6.25118974131979911294601790326, −5.07511229753453376453874336349, −3.80842564834959520680796714861, −3.50514389383045682347497643772, −2.47802105401937259561813189382, −1.30257896553921120174297741290, 0, 1.30257896553921120174297741290, 2.47802105401937259561813189382, 3.50514389383045682347497643772, 3.80842564834959520680796714861, 5.07511229753453376453874336349, 6.25118974131979911294601790326, 6.65095096406869249314131224096, 7.50371880921043911309068608823, 8.315583506182269107091838909957

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