Properties

Label 2-4018-1.1-c1-0-90
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 − 5-s − 6-s − 8-s − 2·9-s + 10-s + 12-s − 2·13-s − 15-s + 16-s + 5·17-s + 2·18-s + 4·19-s − 20-s + 6·23-s − 24-s − 4·25-s + 2·26-s − 5·27-s − 9·29-s + 30-s − 3·31-s − 32-s − 5·34-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.471·18-s + 0.917·19-s − 0.223·20-s + 1.25·23-s − 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.962·27-s − 1.67·29-s + 0.182·30-s − 0.538·31-s − 0.176·32-s − 0.857·34-s − 1/3·36-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 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 7 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 13 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 13 T + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
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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

−8.103708430326373706384192857771, −7.40622866776395561714576772584, −7.10847700689501933078392137011, −5.64666303194085933243943394791, −5.42471489426608259988012070037, −3.96534924163346245201203022127, −3.26336529204343825908177421930, −2.50599759115468772534756961383, −1.35572225109890639211902170084, 0, 1.35572225109890639211902170084, 2.50599759115468772534756961383, 3.26336529204343825908177421930, 3.96534924163346245201203022127, 5.42471489426608259988012070037, 5.64666303194085933243943394791, 7.10847700689501933078392137011, 7.40622866776395561714576772584, 8.103708430326373706384192857771

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