Properties

Label 2-4018-1.1-c1-0-89
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 3·5-s − 6-s + 8-s − 2·9-s − 3·10-s + 4·11-s − 12-s + 2·13-s + 3·15-s + 16-s − 7·17-s − 2·18-s + 4·19-s − 3·20-s + 4·22-s − 4·23-s − 24-s + 4·25-s + 2·26-s + 5·27-s + 9·29-s + 3·30-s + 31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.948·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.774·15-s + 1/4·16-s − 1.69·17-s − 0.471·18-s + 0.917·19-s − 0.670·20-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s + 1.67·29-s + 0.547·30-s + 0.179·31-s + 0.176·32-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: \(1\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 9 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 7 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
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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

−8.077677168247136472411488625880, −7.17906424295495966564164952740, −6.33929390883062526296341267286, −6.11422338306747993709484741313, −4.74058507747248463442445289910, −4.44237954271210154053290210162, −3.54917187090303065610735328827, −2.80319377575133295406951816092, −1.33184807313600885384064697333, 0, 1.33184807313600885384064697333, 2.80319377575133295406951816092, 3.54917187090303065610735328827, 4.44237954271210154053290210162, 4.74058507747248463442445289910, 6.11422338306747993709484741313, 6.33929390883062526296341267286, 7.17906424295495966564164952740, 8.077677168247136472411488625880

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