Properties

Label 2-4018-1.1-c1-0-30
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 − 3.24·3-s + 4-s − 1.45·5-s − 3.24·6-s + 8-s + 7.50·9-s − 1.45·10-s + 4.18·11-s − 3.24·12-s + 5.22·13-s + 4.71·15-s + 16-s + 5.04·17-s + 7.50·18-s − 4.31·19-s − 1.45·20-s + 4.18·22-s − 3.03·23-s − 3.24·24-s − 2.88·25-s + 5.22·26-s − 14.5·27-s + 2.29·29-s + 4.71·30-s + 7.50·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.87·3-s + 0.5·4-s − 0.650·5-s − 1.32·6-s + 0.353·8-s + 2.50·9-s − 0.459·10-s + 1.26·11-s − 0.935·12-s + 1.44·13-s + 1.21·15-s + 0.250·16-s + 1.22·17-s + 1.76·18-s − 0.990·19-s − 0.325·20-s + 0.891·22-s − 0.631·23-s − 0.661·24-s − 0.577·25-s + 1.02·26-s − 2.80·27-s + 0.426·29-s + 0.860·30-s + 1.34·31-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

\(L(1)\) \(\approx\) \(1.682300597\)
\(L(\frac12)\) \(\approx\) \(1.682300597\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.24T + 3T^{2} \)
5 \( 1 + 1.45T + 5T^{2} \)
11 \( 1 - 4.18T + 11T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 - 5.04T + 17T^{2} \)
19 \( 1 + 4.31T + 19T^{2} \)
23 \( 1 + 3.03T + 23T^{2} \)
29 \( 1 - 2.29T + 29T^{2} \)
31 \( 1 - 7.50T + 31T^{2} \)
37 \( 1 - 2.48T + 37T^{2} \)
43 \( 1 - 9.36T + 43T^{2} \)
47 \( 1 + 7.57T + 47T^{2} \)
53 \( 1 - 1.22T + 53T^{2} \)
59 \( 1 + 2.55T + 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 + 8.47T + 71T^{2} \)
73 \( 1 - 4.49T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 2.86T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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

−8.190905817644526112187369085238, −7.48676414864452397182749098541, −6.49512060926102484687720305289, −6.19185927462058949067200505754, −5.68513350745435948539909432297, −4.51493842755200361730051117373, −4.20026162940813773983537957816, −3.39417425863416918767212130239, −1.61622978198534610883606024422, −0.800370533483399141464316645120, 0.800370533483399141464316645120, 1.61622978198534610883606024422, 3.39417425863416918767212130239, 4.20026162940813773983537957816, 4.51493842755200361730051117373, 5.68513350745435948539909432297, 6.19185927462058949067200505754, 6.49512060926102484687720305289, 7.48676414864452397182749098541, 8.190905817644526112187369085238

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