Properties

Label 2-4018-1.1-c1-0-69
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 + 2.95·3-s + 4-s + 2.26·5-s − 2.95·6-s − 8-s + 5.70·9-s − 2.26·10-s − 3.09·11-s + 2.95·12-s + 0.322·13-s + 6.68·15-s + 16-s + 2.44·17-s − 5.70·18-s + 4.20·19-s + 2.26·20-s + 3.09·22-s + 8.85·23-s − 2.95·24-s + 0.128·25-s − 0.322·26-s + 7.99·27-s + 1.00·29-s − 6.68·30-s + 5.72·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.70·3-s + 0.5·4-s + 1.01·5-s − 1.20·6-s − 0.353·8-s + 1.90·9-s − 0.716·10-s − 0.932·11-s + 0.851·12-s + 0.0893·13-s + 1.72·15-s + 0.250·16-s + 0.593·17-s − 1.34·18-s + 0.964·19-s + 0.506·20-s + 0.659·22-s + 1.84·23-s − 0.602·24-s + 0.0256·25-s − 0.0632·26-s + 1.53·27-s + 0.187·29-s − 1.22·30-s + 1.02·31-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

\(L(1)\) \(\approx\) \(3.376850522\)
\(L(\frac12)\) \(\approx\) \(3.376850522\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.95T + 3T^{2} \)
5 \( 1 - 2.26T + 5T^{2} \)
11 \( 1 + 3.09T + 11T^{2} \)
13 \( 1 - 0.322T + 13T^{2} \)
17 \( 1 - 2.44T + 17T^{2} \)
19 \( 1 - 4.20T + 19T^{2} \)
23 \( 1 - 8.85T + 23T^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 - 5.72T + 31T^{2} \)
37 \( 1 + 6.80T + 37T^{2} \)
43 \( 1 - 0.700T + 43T^{2} \)
47 \( 1 + 0.0829T + 47T^{2} \)
53 \( 1 + 6.32T + 53T^{2} \)
59 \( 1 + 9.98T + 59T^{2} \)
61 \( 1 - 2.40T + 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 8.34T + 73T^{2} \)
79 \( 1 - 2.11T + 79T^{2} \)
83 \( 1 + 0.591T + 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 - 4.04T + 97T^{2} \)
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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

−8.604119762409480026576531406211, −7.74753226788222632467817026356, −7.40725583869427183004295222964, −6.47213240945216210593692233926, −5.49625868124313132192652168090, −4.67808119128558903526542614642, −3.22450505669254195218854349210, −2.95399799938444344273798013406, −2.01517167185440915650878801118, −1.17678287563820151936338581690, 1.17678287563820151936338581690, 2.01517167185440915650878801118, 2.95399799938444344273798013406, 3.22450505669254195218854349210, 4.67808119128558903526542614642, 5.49625868124313132192652168090, 6.47213240945216210593692233926, 7.40725583869427183004295222964, 7.74753226788222632467817026356, 8.604119762409480026576531406211

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