Properties

Label 2-8018-1.1-c1-0-49
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 − 1.98·3-s + 4-s − 2.70·5-s + 1.98·6-s + 2.97·7-s − 8-s + 0.924·9-s + 2.70·10-s − 1.51·11-s − 1.98·12-s + 5.78·13-s − 2.97·14-s + 5.34·15-s + 16-s − 3.10·17-s − 0.924·18-s + 19-s − 2.70·20-s − 5.88·21-s + 1.51·22-s − 4.86·23-s + 1.98·24-s + 2.29·25-s − 5.78·26-s + 4.11·27-s + 2.97·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.14·3-s + 0.5·4-s − 1.20·5-s + 0.808·6-s + 1.12·7-s − 0.353·8-s + 0.308·9-s + 0.853·10-s − 0.457·11-s − 0.571·12-s + 1.60·13-s − 0.793·14-s + 1.38·15-s + 0.250·16-s − 0.754·17-s − 0.217·18-s + 0.229·19-s − 0.603·20-s − 1.28·21-s + 0.323·22-s − 1.01·23-s + 0.404·24-s + 0.458·25-s − 1.13·26-s + 0.791·27-s + 0.561·28-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

\(L(1)\) \(\approx\) \(0.6365854659\)
\(L(\frac12)\) \(\approx\) \(0.6365854659\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.98T + 3T^{2} \)
5 \( 1 + 2.70T + 5T^{2} \)
7 \( 1 - 2.97T + 7T^{2} \)
11 \( 1 + 1.51T + 11T^{2} \)
13 \( 1 - 5.78T + 13T^{2} \)
17 \( 1 + 3.10T + 17T^{2} \)
23 \( 1 + 4.86T + 23T^{2} \)
29 \( 1 + 1.87T + 29T^{2} \)
31 \( 1 - 0.914T + 31T^{2} \)
37 \( 1 - 8.04T + 37T^{2} \)
41 \( 1 - 7.68T + 41T^{2} \)
43 \( 1 - 5.81T + 43T^{2} \)
47 \( 1 + 5.88T + 47T^{2} \)
53 \( 1 + 4.81T + 53T^{2} \)
59 \( 1 - 11.4T + 59T^{2} \)
61 \( 1 + 5.49T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 - 6.76T + 73T^{2} \)
79 \( 1 + 3.16T + 79T^{2} \)
83 \( 1 - 1.66T + 83T^{2} \)
89 \( 1 + 5.68T + 89T^{2} \)
97 \( 1 + 5.64T + 97T^{2} \)
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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

−7.954542342987462290843299119354, −7.33466115235161149526741125259, −6.33394490667937315725228629476, −5.96030061987859328420466453269, −5.06816861610080579450043726912, −4.32876244023697190732108886342, −3.69731141837021505408223380973, −2.49316618767106413809850410941, −1.36366797531010831778702018407, −0.51266045314011340134580379853, 0.51266045314011340134580379853, 1.36366797531010831778702018407, 2.49316618767106413809850410941, 3.69731141837021505408223380973, 4.32876244023697190732108886342, 5.06816861610080579450043726912, 5.96030061987859328420466453269, 6.33394490667937315725228629476, 7.33466115235161149526741125259, 7.954542342987462290843299119354

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