Properties

Label 2-4026-1.1-c1-0-27
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 − 3.80·5-s + 6-s − 1.13·7-s − 8-s + 9-s + 3.80·10-s − 11-s − 12-s − 4.24·13-s + 1.13·14-s + 3.80·15-s + 16-s − 0.128·17-s − 18-s − 0.265·19-s − 3.80·20-s + 1.13·21-s + 22-s + 4.37·23-s + 24-s + 9.46·25-s + 4.24·26-s − 27-s − 1.13·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.70·5-s + 0.408·6-s − 0.427·7-s − 0.353·8-s + 0.333·9-s + 1.20·10-s − 0.301·11-s − 0.288·12-s − 1.17·13-s + 0.302·14-s + 0.982·15-s + 0.250·16-s − 0.0312·17-s − 0.235·18-s − 0.0608·19-s − 0.850·20-s + 0.246·21-s + 0.213·22-s + 0.911·23-s + 0.204·24-s + 1.89·25-s + 0.831·26-s − 0.192·27-s − 0.213·28-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 \( 1 + 3.80T + 5T^{2} \)
7 \( 1 + 1.13T + 7T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 0.128T + 17T^{2} \)
19 \( 1 + 0.265T + 19T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + 3.55T + 29T^{2} \)
31 \( 1 - 7.85T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 2.09T + 41T^{2} \)
43 \( 1 - 1.99T + 43T^{2} \)
47 \( 1 + 0.338T + 47T^{2} \)
53 \( 1 + 2.22T + 53T^{2} \)
59 \( 1 + 4.93T + 59T^{2} \)
67 \( 1 - 13.4T + 67T^{2} \)
71 \( 1 - 2.75T + 71T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 + 4.18T + 79T^{2} \)
83 \( 1 + 5.50T + 83T^{2} \)
89 \( 1 + 0.489T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−7.88779402197748399021872231841, −7.50567509780507348376890118701, −6.85205107760481322103732783116, −6.05677786802725642280557283528, −4.91623983312873391462491722681, −4.37113271142612706951773068996, −3.32476063274637045568928169555, −2.53359398700913601427177158751, −0.893372534040728935528300116652, 0, 0.893372534040728935528300116652, 2.53359398700913601427177158751, 3.32476063274637045568928169555, 4.37113271142612706951773068996, 4.91623983312873391462491722681, 6.05677786802725642280557283528, 6.85205107760481322103732783116, 7.50567509780507348376890118701, 7.88779402197748399021872231841

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