Properties

Label 2-8018-1.1-c1-0-31
Degree $2$
Conductor $8018$
Sign $1$
Analytic cond. $64.0240$
Root an. cond. $8.00150$
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  − 2-s − 1.20·3-s + 4-s + 0.0448·5-s + 1.20·6-s − 2.73·7-s − 8-s − 1.53·9-s − 0.0448·10-s + 2.04·11-s − 1.20·12-s + 1.85·13-s + 2.73·14-s − 0.0542·15-s + 16-s + 1.92·17-s + 1.53·18-s − 19-s + 0.0448·20-s + 3.31·21-s − 2.04·22-s − 2.83·23-s + 1.20·24-s − 4.99·25-s − 1.85·26-s + 5.48·27-s − 2.73·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.698·3-s + 0.5·4-s + 0.0200·5-s + 0.493·6-s − 1.03·7-s − 0.353·8-s − 0.511·9-s − 0.0141·10-s + 0.615·11-s − 0.349·12-s + 0.515·13-s + 0.731·14-s − 0.0140·15-s + 0.250·16-s + 0.466·17-s + 0.362·18-s − 0.229·19-s + 0.0100·20-s + 0.723·21-s − 0.435·22-s − 0.590·23-s + 0.246·24-s − 0.999·25-s − 0.364·26-s + 1.05·27-s − 0.517·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8018\)    =    \(2 \cdot 19 \cdot 211\)
Sign: $1$
Analytic conductor: \(64.0240\)
Root analytic conductor: \(8.00150\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5820626194\)
\(L(\frac12)\) \(\approx\) \(0.5820626194\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 + T \)
211 \( 1 - T \)
good3 \( 1 + 1.20T + 3T^{2} \)
5 \( 1 - 0.0448T + 5T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 1.85T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
23 \( 1 + 2.83T + 23T^{2} \)
29 \( 1 - 6.47T + 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + 2.03T + 37T^{2} \)
41 \( 1 - 4.98T + 41T^{2} \)
43 \( 1 + 0.0680T + 43T^{2} \)
47 \( 1 - 9.41T + 47T^{2} \)
53 \( 1 + 9.40T + 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 + 7.94T + 61T^{2} \)
67 \( 1 - 1.66T + 67T^{2} \)
71 \( 1 + 0.505T + 71T^{2} \)
73 \( 1 - 0.785T + 73T^{2} \)
79 \( 1 + 13.5T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 + 5.36T + 97T^{2} \)
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

−7.86842127511508525968543300761, −7.07606662962461093646538640068, −6.32376885367540155077894281945, −6.03417713496676676718654486671, −5.32965192499966450627569329263, −4.17204937211163430221836374047, −3.44357804957850299081342389637, −2.63413618315273181332446806801, −1.51878791468406794723249669037, −0.44412750540045851463990947679, 0.44412750540045851463990947679, 1.51878791468406794723249669037, 2.63413618315273181332446806801, 3.44357804957850299081342389637, 4.17204937211163430221836374047, 5.32965192499966450627569329263, 6.03417713496676676718654486671, 6.32376885367540155077894281945, 7.07606662962461093646538640068, 7.86842127511508525968543300761

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