Properties

Label 2-4017-1.1-c1-0-33
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 $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.52·2-s − 3-s + 0.319·4-s + 3.31·5-s + 1.52·6-s − 2.60·7-s + 2.55·8-s + 9-s − 5.05·10-s + 0.140·11-s − 0.319·12-s − 13-s + 3.97·14-s − 3.31·15-s − 4.53·16-s − 2.15·17-s − 1.52·18-s + 4.38·19-s + 1.05·20-s + 2.60·21-s − 0.214·22-s − 2.12·23-s − 2.55·24-s + 6.00·25-s + 1.52·26-s − 27-s − 0.833·28-s + ⋯
L(s)  = 1  − 1.07·2-s − 0.577·3-s + 0.159·4-s + 1.48·5-s + 0.621·6-s − 0.986·7-s + 0.904·8-s + 0.333·9-s − 1.59·10-s + 0.0423·11-s − 0.0921·12-s − 0.277·13-s + 1.06·14-s − 0.856·15-s − 1.13·16-s − 0.523·17-s − 0.358·18-s + 1.00·19-s + 0.236·20-s + 0.569·21-s − 0.0456·22-s − 0.442·23-s − 0.522·24-s + 1.20·25-s + 0.298·26-s − 0.192·27-s − 0.157·28-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: \(0\)
Selberg data: \((2,\ 4017,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8223540170\)
\(L(\frac12)\) \(\approx\) \(0.8223540170\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.52T + 2T^{2} \)
5 \( 1 - 3.31T + 5T^{2} \)
7 \( 1 + 2.60T + 7T^{2} \)
11 \( 1 - 0.140T + 11T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + 2.12T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 - 7.58T + 31T^{2} \)
37 \( 1 + 1.73T + 37T^{2} \)
41 \( 1 + 2.28T + 41T^{2} \)
43 \( 1 + 0.693T + 43T^{2} \)
47 \( 1 - 8.82T + 47T^{2} \)
53 \( 1 - 7.11T + 53T^{2} \)
59 \( 1 + 7.96T + 59T^{2} \)
61 \( 1 + 0.855T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 3.55T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 0.381T + 83T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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

−8.715573289527417359889040556400, −7.73000636337448175495262885576, −6.93993327263684540178083406536, −6.35825255815963639587530475599, −5.59921204930299540754792599207, −4.93354652748176824074227401484, −3.83992300333750789099628874173, −2.58768780317264639181225220958, −1.71176142290268763950917555167, −0.64168133391442753774688476664, 0.64168133391442753774688476664, 1.71176142290268763950917555167, 2.58768780317264639181225220958, 3.83992300333750789099628874173, 4.93354652748176824074227401484, 5.59921204930299540754792599207, 6.35825255815963639587530475599, 6.93993327263684540178083406536, 7.73000636337448175495262885576, 8.715573289527417359889040556400

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