Properties

Label 2-4017-1.1-c1-0-160
Degree $2$
Conductor $4017$
Sign $-1$
Analytic cond. $32.0759$
Root an. cond. $5.66355$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.820·2-s + 3-s − 1.32·4-s + 3.24·5-s − 0.820·6-s − 1.69·7-s + 2.72·8-s + 9-s − 2.66·10-s − 2.47·11-s − 1.32·12-s − 13-s + 1.39·14-s + 3.24·15-s + 0.413·16-s − 0.439·17-s − 0.820·18-s − 2.19·19-s − 4.31·20-s − 1.69·21-s + 2.02·22-s + 0.232·23-s + 2.72·24-s + 5.56·25-s + 0.820·26-s + 27-s + 2.25·28-s + ⋯
L(s)  = 1  − 0.580·2-s + 0.577·3-s − 0.663·4-s + 1.45·5-s − 0.334·6-s − 0.641·7-s + 0.965·8-s + 0.333·9-s − 0.843·10-s − 0.745·11-s − 0.382·12-s − 0.277·13-s + 0.372·14-s + 0.839·15-s + 0.103·16-s − 0.106·17-s − 0.193·18-s − 0.503·19-s − 0.964·20-s − 0.370·21-s + 0.432·22-s + 0.0485·23-s + 0.557·24-s + 1.11·25-s + 0.160·26-s + 0.192·27-s + 0.425·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4017\)    =    \(3 \cdot 13 \cdot 103\)
Sign: $-1$
Analytic conductor: \(32.0759\)
Root analytic conductor: \(5.66355\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4017,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
13 \( 1 + T \)
103 \( 1 - T \)
good2 \( 1 + 0.820T + 2T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
7 \( 1 + 1.69T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
17 \( 1 + 0.439T + 17T^{2} \)
19 \( 1 + 2.19T + 19T^{2} \)
23 \( 1 - 0.232T + 23T^{2} \)
29 \( 1 + 3.67T + 29T^{2} \)
31 \( 1 + 7.98T + 31T^{2} \)
37 \( 1 - 6.76T + 37T^{2} \)
41 \( 1 - 2.73T + 41T^{2} \)
43 \( 1 + 6.45T + 43T^{2} \)
47 \( 1 - 4.09T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 8.89T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 2.29T + 67T^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 - 3.40T + 73T^{2} \)
79 \( 1 - 1.40T + 79T^{2} \)
83 \( 1 + 6.57T + 83T^{2} \)
89 \( 1 - 9.38T + 89T^{2} \)
97 \( 1 + 14.6T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.185980040180968042872139593129, −7.55021285454094690162091493312, −6.67503649274720224558160246803, −5.82322730576249040805715574420, −5.16991155733682109049716116759, −4.28521110284215527474389191569, −3.23959877975180723445633320122, −2.30836396127493035413230815038, −1.51229357085470165655104196778, 0, 1.51229357085470165655104196778, 2.30836396127493035413230815038, 3.23959877975180723445633320122, 4.28521110284215527474389191569, 5.16991155733682109049716116759, 5.82322730576249040805715574420, 6.67503649274720224558160246803, 7.55021285454094690162091493312, 8.185980040180968042872139593129

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