Properties

Label 2-4026-1.1-c1-0-76
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 − 0.0775·5-s + 6-s + 2.74·7-s − 8-s + 9-s + 0.0775·10-s − 11-s − 12-s + 4.76·13-s − 2.74·14-s + 0.0775·15-s + 16-s + 4.60·17-s − 18-s − 5.24·19-s − 0.0775·20-s − 2.74·21-s + 22-s − 9.37·23-s + 24-s − 4.99·25-s − 4.76·26-s − 27-s + 2.74·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0346·5-s + 0.408·6-s + 1.03·7-s − 0.353·8-s + 0.333·9-s + 0.0245·10-s − 0.301·11-s − 0.288·12-s + 1.32·13-s − 0.732·14-s + 0.0200·15-s + 0.250·16-s + 1.11·17-s − 0.235·18-s − 1.20·19-s − 0.0173·20-s − 0.598·21-s + 0.213·22-s − 1.95·23-s + 0.204·24-s − 0.998·25-s − 0.935·26-s − 0.192·27-s + 0.518·28-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 \( 1 + 0.0775T + 5T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
13 \( 1 - 4.76T + 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 + 5.24T + 19T^{2} \)
23 \( 1 + 9.37T + 23T^{2} \)
29 \( 1 + 4.01T + 29T^{2} \)
31 \( 1 - 3.80T + 31T^{2} \)
37 \( 1 + 0.483T + 37T^{2} \)
41 \( 1 - 1.38T + 41T^{2} \)
43 \( 1 + 5.58T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 4.36T + 53T^{2} \)
59 \( 1 + 4.92T + 59T^{2} \)
67 \( 1 + 2.49T + 67T^{2} \)
71 \( 1 - 0.600T + 71T^{2} \)
73 \( 1 - 8.32T + 73T^{2} \)
79 \( 1 + 0.850T + 79T^{2} \)
83 \( 1 + 0.613T + 83T^{2} \)
89 \( 1 - 0.357T + 89T^{2} \)
97 \( 1 - 9.29T + 97T^{2} \)
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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

−8.023130013832405490500967322582, −7.71830792528675282505164142474, −6.44057474464022045062371343668, −6.05041555096889110910775842750, −5.22835227253193985607882928780, −4.25837770274494831191350596390, −3.47691897808482479205335829544, −2.02006765030001935916242400194, −1.41325197757227575605254124140, 0, 1.41325197757227575605254124140, 2.02006765030001935916242400194, 3.47691897808482479205335829544, 4.25837770274494831191350596390, 5.22835227253193985607882928780, 6.05041555096889110910775842750, 6.44057474464022045062371343668, 7.71830792528675282505164142474, 8.023130013832405490500967322582

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