Properties

Label 2-8018-1.1-c1-0-16
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 − 1.99·3-s + 4-s − 0.291·5-s − 1.99·6-s − 0.492·7-s + 8-s + 0.989·9-s − 0.291·10-s − 5.82·11-s − 1.99·12-s − 2.76·13-s − 0.492·14-s + 0.582·15-s + 16-s − 7.53·17-s + 0.989·18-s − 19-s − 0.291·20-s + 0.984·21-s − 5.82·22-s − 1.16·23-s − 1.99·24-s − 4.91·25-s − 2.76·26-s + 4.01·27-s − 0.492·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.130·5-s − 0.815·6-s − 0.186·7-s + 0.353·8-s + 0.329·9-s − 0.0922·10-s − 1.75·11-s − 0.576·12-s − 0.767·13-s − 0.131·14-s + 0.150·15-s + 0.250·16-s − 1.82·17-s + 0.233·18-s − 0.229·19-s − 0.0652·20-s + 0.214·21-s − 1.24·22-s − 0.242·23-s − 0.407·24-s − 0.982·25-s − 0.543·26-s + 0.772·27-s − 0.0931·28-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

\(L(1)\) \(\approx\) \(0.4253880256\)
\(L(\frac12)\) \(\approx\) \(0.4253880256\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.99T + 3T^{2} \)
5 \( 1 + 0.291T + 5T^{2} \)
7 \( 1 + 0.492T + 7T^{2} \)
11 \( 1 + 5.82T + 11T^{2} \)
13 \( 1 + 2.76T + 13T^{2} \)
17 \( 1 + 7.53T + 17T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 + 5.53T + 29T^{2} \)
31 \( 1 - 0.123T + 31T^{2} \)
37 \( 1 - 0.138T + 37T^{2} \)
41 \( 1 + 0.934T + 41T^{2} \)
43 \( 1 - 1.73T + 43T^{2} \)
47 \( 1 - 3.19T + 47T^{2} \)
53 \( 1 - 0.526T + 53T^{2} \)
59 \( 1 + 7.55T + 59T^{2} \)
61 \( 1 - 5.41T + 61T^{2} \)
67 \( 1 + 2.44T + 67T^{2} \)
71 \( 1 + 5.21T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 - 6.09T + 79T^{2} \)
83 \( 1 - 0.696T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 3.77T + 97T^{2} \)
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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

−7.62318576038894219605962911758, −6.99258842461938704217877093863, −6.22305441985093076906185031813, −5.73701658952529792639580886602, −4.95648877094744624672899793481, −4.65219526658633482104722009546, −3.65515128257343456100258964437, −2.58865087604178211610019100795, −2.04840627806191126440309556891, −0.28138274544797456354102337111, 0.28138274544797456354102337111, 2.04840627806191126440309556891, 2.58865087604178211610019100795, 3.65515128257343456100258964437, 4.65219526658633482104722009546, 4.95648877094744624672899793481, 5.73701658952529792639580886602, 6.22305441985093076906185031813, 6.99258842461938704217877093863, 7.62318576038894219605962911758

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