Properties

Label 2-4027-1.1-c1-0-325
Degree $2$
Conductor $4027$
Sign $1$
Analytic cond. $32.1557$
Root an. cond. $5.67060$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·2-s + 1.23·3-s + 0.618·4-s − 5-s − 2.00·6-s − 4.61·7-s + 2.23·8-s − 1.47·9-s + 1.61·10-s − 11-s + 0.763·12-s − 2.61·13-s + 7.47·14-s − 1.23·15-s − 4.85·16-s − 5.09·17-s + 2.38·18-s − 5·19-s − 0.618·20-s − 5.70·21-s + 1.61·22-s − 6.61·23-s + 2.76·24-s − 4·25-s + 4.23·26-s − 5.52·27-s − 2.85·28-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.713·3-s + 0.309·4-s − 0.447·5-s − 0.816·6-s − 1.74·7-s + 0.790·8-s − 0.490·9-s + 0.511·10-s − 0.301·11-s + 0.220·12-s − 0.726·13-s + 1.99·14-s − 0.319·15-s − 1.21·16-s − 1.23·17-s + 0.561·18-s − 1.14·19-s − 0.138·20-s − 1.24·21-s + 0.344·22-s − 1.37·23-s + 0.564·24-s − 0.800·25-s + 0.830·26-s − 1.06·27-s − 0.539·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4027\)
Sign: $1$
Analytic conductor: \(32.1557\)
Root analytic conductor: \(5.67060\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 4027,\ (\ :1/2),\ 1)\)

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)$
bad4027 \( 1+O(T) \)
good2 \( 1 + 1.61T + 2T^{2} \)
3 \( 1 - 1.23T + 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 + 4.61T + 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 2.61T + 13T^{2} \)
17 \( 1 + 5.09T + 17T^{2} \)
19 \( 1 + 5T + 19T^{2} \)
23 \( 1 + 6.61T + 23T^{2} \)
29 \( 1 + 3.38T + 29T^{2} \)
31 \( 1 + 3.76T + 31T^{2} \)
37 \( 1 - 6.94T + 37T^{2} \)
41 \( 1 + 2.76T + 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 - 0.472T + 47T^{2} \)
53 \( 1 - 1.76T + 53T^{2} \)
59 \( 1 - 4.70T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 + 4.14T + 71T^{2} \)
73 \( 1 - 0.0901T + 73T^{2} \)
79 \( 1 - 13T + 79T^{2} \)
83 \( 1 + 16.5T + 83T^{2} \)
89 \( 1 + 4.14T + 89T^{2} \)
97 \( 1 - 4.52T + 97T^{2} \)
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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

−7.85224436871483197335649174826, −7.30974896975348319763846171450, −6.44088221649436062100150352183, −5.74584922460636842827468475644, −4.31717581832264436454681995722, −3.82404021946333422453026596635, −2.67061473860151061007137077991, −2.09389044360463292709620624657, 0, 0, 2.09389044360463292709620624657, 2.67061473860151061007137077991, 3.82404021946333422453026596635, 4.31717581832264436454681995722, 5.74584922460636842827468475644, 6.44088221649436062100150352183, 7.30974896975348319763846171450, 7.85224436871483197335649174826

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