Properties

Label 2-4018-1.1-c1-0-14
Degree $2$
Conductor $4018$
Sign $1$
Analytic cond. $32.0838$
Root an. cond. $5.66426$
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 − 1.63·3-s + 4-s − 2.84·5-s + 1.63·6-s − 8-s − 0.328·9-s + 2.84·10-s + 5.83·11-s − 1.63·12-s + 2.22·13-s + 4.65·15-s + 16-s + 1.51·17-s + 0.328·18-s + 1.82·19-s − 2.84·20-s − 5.83·22-s − 1.53·23-s + 1.63·24-s + 3.10·25-s − 2.22·26-s + 5.44·27-s + 3.43·29-s − 4.65·30-s − 7.63·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.943·3-s + 0.5·4-s − 1.27·5-s + 0.667·6-s − 0.353·8-s − 0.109·9-s + 0.900·10-s + 1.76·11-s − 0.471·12-s + 0.616·13-s + 1.20·15-s + 0.250·16-s + 0.368·17-s + 0.0774·18-s + 0.418·19-s − 0.636·20-s − 1.24·22-s − 0.319·23-s + 0.333·24-s + 0.620·25-s − 0.436·26-s + 1.04·27-s + 0.637·29-s − 0.849·30-s − 1.37·31-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6833069716\)
\(L(\frac12)\) \(\approx\) \(0.6833069716\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.63T + 3T^{2} \)
5 \( 1 + 2.84T + 5T^{2} \)
11 \( 1 - 5.83T + 11T^{2} \)
13 \( 1 - 2.22T + 13T^{2} \)
17 \( 1 - 1.51T + 17T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 + 1.53T + 23T^{2} \)
29 \( 1 - 3.43T + 29T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 - 4.42T + 37T^{2} \)
43 \( 1 - 4.53T + 43T^{2} \)
47 \( 1 + 1.02T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 + 0.382T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 5.14T + 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 + 8.71T + 89T^{2} \)
97 \( 1 - 1.15T + 97T^{2} \)
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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

−8.612873440868585848349833658005, −7.59361813904495610183572590288, −7.13899775301127120443148204807, −6.20583183987359145438688475279, −5.80572587186550554776281544054, −4.56608210299385663424737799084, −3.87452391278917192512553832077, −3.08234856697033809709450643040, −1.48222195037356589258050387430, −0.59714402035663802350516614070, 0.59714402035663802350516614070, 1.48222195037356589258050387430, 3.08234856697033809709450643040, 3.87452391278917192512553832077, 4.56608210299385663424737799084, 5.80572587186550554776281544054, 6.20583183987359145438688475279, 7.13899775301127120443148204807, 7.59361813904495610183572590288, 8.612873440868585848349833658005

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