Properties

Label 2-4018-1.1-c1-0-118
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 + 2.41·3-s + 4-s + 3.82·5-s + 2.41·6-s + 8-s + 2.82·9-s + 3.82·10-s + 2.24·11-s + 2.41·12-s + 3.41·13-s + 9.24·15-s + 16-s − 1.82·17-s + 2.82·18-s − 2.82·19-s + 3.82·20-s + 2.24·22-s − 5.65·23-s + 2.41·24-s + 9.65·25-s + 3.41·26-s − 0.414·27-s − 5·29-s + 9.24·30-s − 8.41·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.39·3-s + 0.5·4-s + 1.71·5-s + 0.985·6-s + 0.353·8-s + 0.942·9-s + 1.21·10-s + 0.676·11-s + 0.696·12-s + 0.946·13-s + 2.38·15-s + 0.250·16-s − 0.443·17-s + 0.666·18-s − 0.648·19-s + 0.856·20-s + 0.478·22-s − 1.17·23-s + 0.492·24-s + 1.93·25-s + 0.669·26-s − 0.0797·27-s − 0.928·29-s + 1.68·30-s − 1.51·31-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

\(L(1)\) \(\approx\) \(7.026339463\)
\(L(\frac12)\) \(\approx\) \(7.026339463\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 - 3.82T + 5T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 + 1.82T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 8.41T + 31T^{2} \)
37 \( 1 - 9.41T + 37T^{2} \)
43 \( 1 + 7.24T + 43T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + 6.48T + 59T^{2} \)
61 \( 1 + 5.48T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 0.757T + 79T^{2} \)
83 \( 1 - 5.75T + 83T^{2} \)
89 \( 1 + 1.82T + 89T^{2} \)
97 \( 1 - 5.82T + 97T^{2} \)
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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

−8.644907930985845582965819274730, −7.72942534799284651813487818808, −6.86123036651845838446535259478, −6.01327976419981801911694296325, −5.74869129574092192796578771888, −4.46304335321314618777407361582, −3.76093908948347490920581515469, −2.92310639294324702810369196449, −1.99748729167342234837029981045, −1.64982416714107167515857975677, 1.64982416714107167515857975677, 1.99748729167342234837029981045, 2.92310639294324702810369196449, 3.76093908948347490920581515469, 4.46304335321314618777407361582, 5.74869129574092192796578771888, 6.01327976419981801911694296325, 6.86123036651845838446535259478, 7.72942534799284651813487818808, 8.644907930985845582965819274730

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