Properties

Label 2-4026-1.1-c1-0-53
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 − 3.39·5-s − 6-s − 4.14·7-s + 8-s + 9-s − 3.39·10-s + 11-s − 12-s + 3.54·13-s − 4.14·14-s + 3.39·15-s + 16-s + 17-s + 18-s + 6.54·19-s − 3.39·20-s + 4.14·21-s + 22-s + 4.54·23-s − 24-s + 6.54·25-s + 3.54·26-s − 27-s − 4.14·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.51·5-s − 0.408·6-s − 1.56·7-s + 0.353·8-s + 0.333·9-s − 1.07·10-s + 0.301·11-s − 0.288·12-s + 0.982·13-s − 1.10·14-s + 0.877·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.50·19-s − 0.759·20-s + 0.904·21-s + 0.213·22-s + 0.947·23-s − 0.204·24-s + 1.30·25-s + 0.695·26-s − 0.192·27-s − 0.783·28-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 \( 1 + 3.39T + 5T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
13 \( 1 - 3.54T + 13T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 - 6.54T + 19T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + 9.14T + 31T^{2} \)
37 \( 1 + 5.74T + 37T^{2} \)
41 \( 1 - 0.497T + 41T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + 1.90T + 47T^{2} \)
53 \( 1 + 6.69T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
67 \( 1 - 9.14T + 67T^{2} \)
71 \( 1 + 9.14T + 71T^{2} \)
73 \( 1 - 4.64T + 73T^{2} \)
79 \( 1 - 3.54T + 79T^{2} \)
83 \( 1 + 17.7T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−7.82773311101280739098928041196, −7.04483914453183360312576505885, −6.75825759159854996255530425037, −5.75435964542530765566829335260, −5.15821540174972337854022879804, −3.99703069694419138957501151629, −3.56221318610584867510695219703, −3.02675117602580223500398283999, −1.21635534741111314794764307812, 0, 1.21635534741111314794764307812, 3.02675117602580223500398283999, 3.56221318610584867510695219703, 3.99703069694419138957501151629, 5.15821540174972337854022879804, 5.75435964542530765566829335260, 6.75825759159854996255530425037, 7.04483914453183360312576505885, 7.82773311101280739098928041196

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