Properties

Label 2-4026-1.1-c1-0-61
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 − 2.87·5-s − 6-s − 0.347·7-s − 8-s + 9-s + 2.87·10-s + 11-s + 12-s − 2.53·13-s + 0.347·14-s − 2.87·15-s + 16-s + 4.75·17-s − 18-s − 2.83·19-s − 2.87·20-s − 0.347·21-s − 22-s − 3.53·23-s − 24-s + 3.29·25-s + 2.53·26-s + 27-s − 0.347·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.28·5-s − 0.408·6-s − 0.131·7-s − 0.353·8-s + 0.333·9-s + 0.910·10-s + 0.301·11-s + 0.288·12-s − 0.702·13-s + 0.0928·14-s − 0.743·15-s + 0.250·16-s + 1.15·17-s − 0.235·18-s − 0.650·19-s − 0.643·20-s − 0.0757·21-s − 0.213·22-s − 0.736·23-s − 0.204·24-s + 0.658·25-s + 0.496·26-s + 0.192·27-s − 0.0656·28-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 \( 1 + 2.87T + 5T^{2} \)
7 \( 1 + 0.347T + 7T^{2} \)
13 \( 1 + 2.53T + 13T^{2} \)
17 \( 1 - 4.75T + 17T^{2} \)
19 \( 1 + 2.83T + 19T^{2} \)
23 \( 1 + 3.53T + 23T^{2} \)
29 \( 1 - 8.47T + 29T^{2} \)
31 \( 1 - 2.17T + 31T^{2} \)
37 \( 1 - 1.90T + 37T^{2} \)
41 \( 1 + 1.24T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 2.20T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + 0.781T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 0.532T + 79T^{2} \)
83 \( 1 + 9.62T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 7.80T + 97T^{2} \)
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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

−8.026021737622905351765329021845, −7.66263901585514643269519273924, −6.87163496228250944200134569512, −6.12388651727663782495362409260, −4.89648016076150829914454199365, −4.10608850285677161827857621597, −3.31746901204759266398855489828, −2.52881656452819256947787415540, −1.26428425200625308885363728382, 0, 1.26428425200625308885363728382, 2.52881656452819256947787415540, 3.31746901204759266398855489828, 4.10608850285677161827857621597, 4.89648016076150829914454199365, 6.12388651727663782495362409260, 6.87163496228250944200134569512, 7.66263901585514643269519273924, 8.026021737622905351765329021845

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