Properties

Label 2-4018-1.1-c1-0-74
Degree $2$
Conductor $4018$
Sign $-1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
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 − 2·3-s + 4-s − 2.73·5-s − 2·6-s + 8-s + 9-s − 2.73·10-s − 2.73·11-s − 2·12-s + 5.46·13-s + 5.46·15-s + 16-s + 18-s + 3.46·19-s − 2.73·20-s − 2.73·22-s − 2·23-s − 2·24-s + 2.46·25-s + 5.46·26-s + 4·27-s − 7.66·29-s + 5.46·30-s − 1.46·31-s + 32-s + 5.46·33-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.5·4-s − 1.22·5-s − 0.816·6-s + 0.353·8-s + 0.333·9-s − 0.863·10-s − 0.823·11-s − 0.577·12-s + 1.51·13-s + 1.41·15-s + 0.250·16-s + 0.235·18-s + 0.794·19-s − 0.610·20-s − 0.582·22-s − 0.417·23-s − 0.408·24-s + 0.492·25-s + 1.07·26-s + 0.769·27-s − 1.42·29-s + 0.997·30-s − 0.262·31-s + 0.176·32-s + 0.951·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
Sign: $-1$
Analytic conductor: \(32.0838\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 2T + 3T^{2} \)
5 \( 1 + 2.73T + 5T^{2} \)
11 \( 1 + 2.73T + 11T^{2} \)
13 \( 1 - 5.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 3.46T + 19T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 7.66T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 4.92T + 37T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + 0.535T + 47T^{2} \)
53 \( 1 - 7.66T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 + 6.73T + 61T^{2} \)
67 \( 1 - 0.196T + 67T^{2} \)
71 \( 1 - 2.53T + 71T^{2} \)
73 \( 1 - 8.92T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 12.9T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−7.79592827273602933833804996934, −7.37049211954217764058573524974, −6.34539695193228943783254383080, −5.76236492837864860855716988318, −5.20788284114810811986425276123, −4.21803941925655274431191779851, −3.70982137288922512259367674243, −2.73257066438540463357652840204, −1.20760190856188987621636981800, 0, 1.20760190856188987621636981800, 2.73257066438540463357652840204, 3.70982137288922512259367674243, 4.21803941925655274431191779851, 5.20788284114810811986425276123, 5.76236492837864860855716988318, 6.34539695193228943783254383080, 7.37049211954217764058573524974, 7.79592827273602933833804996934

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