Properties

Label 2-4018-1.1-c1-0-4
Degree $2$
Conductor $4018$
Sign $1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 0.757·3-s + 4-s − 1.30·5-s + 0.757·6-s − 8-s − 2.42·9-s + 1.30·10-s + 1.45·11-s − 0.757·12-s − 3.28·13-s + 0.989·15-s + 16-s − 7.19·17-s + 2.42·18-s − 5.28·19-s − 1.30·20-s − 1.45·22-s − 3.28·23-s + 0.757·24-s − 3.29·25-s + 3.28·26-s + 4.10·27-s + 6.46·29-s − 0.989·30-s − 3.01·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.437·3-s + 0.5·4-s − 0.584·5-s + 0.309·6-s − 0.353·8-s − 0.808·9-s + 0.413·10-s + 0.437·11-s − 0.218·12-s − 0.910·13-s + 0.255·15-s + 0.250·16-s − 1.74·17-s + 0.571·18-s − 1.21·19-s − 0.292·20-s − 0.309·22-s − 0.684·23-s + 0.154·24-s − 0.658·25-s + 0.643·26-s + 0.790·27-s + 1.19·29-s − 0.180·30-s − 0.540·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
Sign: $1$
Analytic conductor: \(32.0838\)
Root analytic conductor: \(5.66426\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2971312214\)
\(L(\frac12)\) \(\approx\) \(0.2971312214\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + 0.757T + 3T^{2} \)
5 \( 1 + 1.30T + 5T^{2} \)
11 \( 1 - 1.45T + 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + 7.19T + 17T^{2} \)
19 \( 1 + 5.28T + 19T^{2} \)
23 \( 1 + 3.28T + 23T^{2} \)
29 \( 1 - 6.46T + 29T^{2} \)
31 \( 1 + 3.01T + 31T^{2} \)
37 \( 1 + 3.51T + 37T^{2} \)
43 \( 1 - 0.525T + 43T^{2} \)
47 \( 1 + 3.15T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 1.68T + 59T^{2} \)
61 \( 1 - 0.842T + 61T^{2} \)
67 \( 1 - 7.88T + 67T^{2} \)
71 \( 1 + 16.4T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 0.168T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 3.62T + 97T^{2} \)
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

−8.593795925745252983628941589844, −7.83297529628379447791154242557, −6.89752075000561427996918801577, −6.48064114537869426720774259984, −5.61304250300732347965019361747, −4.62032592422151729768338473786, −3.93469637650030537118840728836, −2.70172766854052546479075477054, −1.95890264744611950581604308450, −0.33251683220817613587042227718, 0.33251683220817613587042227718, 1.95890264744611950581604308450, 2.70172766854052546479075477054, 3.93469637650030537118840728836, 4.62032592422151729768338473786, 5.61304250300732347965019361747, 6.48064114537869426720774259984, 6.89752075000561427996918801577, 7.83297529628379447791154242557, 8.593795925745252983628941589844

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