Properties

Label 2-8018-1.1-c1-0-269
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.32·3-s + 4-s + 1.93·5-s + 2.32·6-s + 2.88·7-s + 8-s + 2.40·9-s + 1.93·10-s + 1.33·11-s + 2.32·12-s + 2.89·13-s + 2.88·14-s + 4.50·15-s + 16-s + 5.30·17-s + 2.40·18-s + 19-s + 1.93·20-s + 6.70·21-s + 1.33·22-s − 5.19·23-s + 2.32·24-s − 1.24·25-s + 2.89·26-s − 1.37·27-s + 2.88·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.34·3-s + 0.5·4-s + 0.866·5-s + 0.949·6-s + 1.08·7-s + 0.353·8-s + 0.803·9-s + 0.612·10-s + 0.402·11-s + 0.671·12-s + 0.803·13-s + 0.770·14-s + 1.16·15-s + 0.250·16-s + 1.28·17-s + 0.567·18-s + 0.229·19-s + 0.433·20-s + 1.46·21-s + 0.284·22-s − 1.08·23-s + 0.474·24-s − 0.249·25-s + 0.568·26-s − 0.264·27-s + 0.544·28-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

\(L(1)\) \(\approx\) \(8.087533108\)
\(L(\frac12)\) \(\approx\) \(8.087533108\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.32T + 3T^{2} \)
5 \( 1 - 1.93T + 5T^{2} \)
7 \( 1 - 2.88T + 7T^{2} \)
11 \( 1 - 1.33T + 11T^{2} \)
13 \( 1 - 2.89T + 13T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
23 \( 1 + 5.19T + 23T^{2} \)
29 \( 1 + 3.02T + 29T^{2} \)
31 \( 1 - 5.59T + 31T^{2} \)
37 \( 1 + 5.56T + 37T^{2} \)
41 \( 1 + 2.65T + 41T^{2} \)
43 \( 1 - 0.276T + 43T^{2} \)
47 \( 1 - 0.439T + 47T^{2} \)
53 \( 1 - 7.22T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 0.972T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 + 1.73T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
97 \( 1 + 3.33T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.82496742615151635450523039611, −7.37117956687622992281837424954, −6.19796618159526998939112681439, −5.81191437930284966864913798293, −4.97532776804384379588627288486, −4.13942642907152369770854197180, −3.47205897637348550589144741155, −2.77579671478030625618154193281, −1.77162124680392699471205056025, −1.47902558839117623134852902831, 1.47902558839117623134852902831, 1.77162124680392699471205056025, 2.77579671478030625618154193281, 3.47205897637348550589144741155, 4.13942642907152369770854197180, 4.97532776804384379588627288486, 5.81191437930284966864913798293, 6.19796618159526998939112681439, 7.37117956687622992281837424954, 7.82496742615151635450523039611

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