Properties

Label 2-4018-1.1-c1-0-72
Degree $2$
Conductor $4018$
Sign $1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 4·5-s + 8-s − 3·9-s + 4·10-s − 2·11-s + 6·13-s + 16-s + 6·17-s − 3·18-s − 4·19-s + 4·20-s − 2·22-s + 8·23-s + 11·25-s + 6·26-s − 8·29-s + 32-s + 6·34-s − 3·36-s − 2·37-s − 4·38-s + 4·40-s + 41-s − 8·43-s − 2·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s − 9-s + 1.26·10-s − 0.603·11-s + 1.66·13-s + 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.894·20-s − 0.426·22-s + 1.66·23-s + 11/5·25-s + 1.17·26-s − 1.48·29-s + 0.176·32-s + 1.02·34-s − 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.632·40-s + 0.156·41-s − 1.21·43-s − 0.301·44-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.433927792\)
\(L(\frac12)\) \(\approx\) \(4.433927792\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.613816602334229686054100646789, −7.65721528955671107769156120958, −6.56631387530868863503842917299, −6.11127408256228484107874823281, −5.40935598844077101472264620739, −5.12721343255384048228575572204, −3.64099149589075833458053763761, −2.99944773900680661378778345491, −2.09567390840630364723470427188, −1.18690792211090642717452721795, 1.18690792211090642717452721795, 2.09567390840630364723470427188, 2.99944773900680661378778345491, 3.64099149589075833458053763761, 5.12721343255384048228575572204, 5.40935598844077101472264620739, 6.11127408256228484107874823281, 6.56631387530868863503842917299, 7.65721528955671107769156120958, 8.613816602334229686054100646789

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