Properties

Label 2-4028-1.1-c1-0-70
Degree $2$
Conductor $4028$
Sign $-1$
Analytic cond. $32.1637$
Root an. cond. $5.67130$
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  + 1.49·3-s − 0.450·5-s + 1.77·7-s − 0.777·9-s − 2.05·11-s + 1.50·13-s − 0.671·15-s − 1.71·17-s + 19-s + 2.64·21-s − 3.69·23-s − 4.79·25-s − 5.63·27-s − 6.49·29-s − 5.66·31-s − 3.06·33-s − 0.798·35-s − 6.57·37-s + 2.24·39-s + 11.3·41-s − 6.96·43-s + 0.350·45-s − 7.28·47-s − 3.85·49-s − 2.55·51-s − 53-s + 0.925·55-s + ⋯
L(s)  = 1  + 0.860·3-s − 0.201·5-s + 0.670·7-s − 0.259·9-s − 0.619·11-s + 0.417·13-s − 0.173·15-s − 0.416·17-s + 0.229·19-s + 0.576·21-s − 0.769·23-s − 0.959·25-s − 1.08·27-s − 1.20·29-s − 1.01·31-s − 0.532·33-s − 0.135·35-s − 1.08·37-s + 0.359·39-s + 1.76·41-s − 1.06·43-s + 0.0522·45-s − 1.06·47-s − 0.550·49-s − 0.358·51-s − 0.137·53-s + 0.124·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4028\)    =    \(2^{2} \cdot 19 \cdot 53\)
Sign: $-1$
Analytic conductor: \(32.1637\)
Root analytic conductor: \(5.67130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4028,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
53 \( 1 + T \)
good3 \( 1 - 1.49T + 3T^{2} \)
5 \( 1 + 0.450T + 5T^{2} \)
7 \( 1 - 1.77T + 7T^{2} \)
11 \( 1 + 2.05T + 11T^{2} \)
13 \( 1 - 1.50T + 13T^{2} \)
17 \( 1 + 1.71T + 17T^{2} \)
23 \( 1 + 3.69T + 23T^{2} \)
29 \( 1 + 6.49T + 29T^{2} \)
31 \( 1 + 5.66T + 31T^{2} \)
37 \( 1 + 6.57T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 6.96T + 43T^{2} \)
47 \( 1 + 7.28T + 47T^{2} \)
59 \( 1 - 5.06T + 59T^{2} \)
61 \( 1 - 8.91T + 61T^{2} \)
67 \( 1 - 9.61T + 67T^{2} \)
71 \( 1 - 15.9T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 - 7.87T + 79T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 - 14.2T + 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.014835310254981874714538178955, −7.70719906595987933481884616276, −6.73146306253549005129899195848, −5.69763132518136904599745963590, −5.16751057623589874466337657920, −4.00207916114035968313246008408, −3.49762702360845585030742615814, −2.38720339634884351982593872412, −1.73324939916984053669346711452, 0, 1.73324939916984053669346711452, 2.38720339634884351982593872412, 3.49762702360845585030742615814, 4.00207916114035968313246008408, 5.16751057623589874466337657920, 5.69763132518136904599745963590, 6.73146306253549005129899195848, 7.70719906595987933481884616276, 8.014835310254981874714538178955

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