Properties

Label 2-4018-1.1-c1-0-9
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 − 2.71·3-s + 4-s − 0.432·5-s − 2.71·6-s + 8-s + 4.34·9-s − 0.432·10-s − 4.46·11-s − 2.71·12-s − 6.22·13-s + 1.17·15-s + 16-s − 3.78·17-s + 4.34·18-s + 2.18·19-s − 0.432·20-s − 4.46·22-s − 0.639·23-s − 2.71·24-s − 4.81·25-s − 6.22·26-s − 3.65·27-s + 3.86·29-s + 1.17·30-s + 0.411·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.56·3-s + 0.5·4-s − 0.193·5-s − 1.10·6-s + 0.353·8-s + 1.44·9-s − 0.136·10-s − 1.34·11-s − 0.782·12-s − 1.72·13-s + 0.302·15-s + 0.250·16-s − 0.917·17-s + 1.02·18-s + 0.501·19-s − 0.0967·20-s − 0.951·22-s − 0.133·23-s − 0.553·24-s − 0.962·25-s − 1.22·26-s − 0.704·27-s + 0.718·29-s + 0.214·30-s + 0.0738·31-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

\(L(1)\) \(\approx\) \(0.8273108037\)
\(L(\frac12)\) \(\approx\) \(0.8273108037\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 + 0.432T + 5T^{2} \)
11 \( 1 + 4.46T + 11T^{2} \)
13 \( 1 + 6.22T + 13T^{2} \)
17 \( 1 + 3.78T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 + 0.639T + 23T^{2} \)
29 \( 1 - 3.86T + 29T^{2} \)
31 \( 1 - 0.411T + 31T^{2} \)
37 \( 1 - 7.45T + 37T^{2} \)
43 \( 1 - 2.67T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 + 5.53T + 67T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 - 6.26T + 73T^{2} \)
79 \( 1 + 0.703T + 79T^{2} \)
83 \( 1 - 10.8T + 83T^{2} \)
89 \( 1 - 1.59T + 89T^{2} \)
97 \( 1 - 15.6T + 97T^{2} \)
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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

−8.093321580282230601036084586032, −7.47392474996511657289131602222, −6.83204491570017414698661590457, −6.04544470125185312587819187384, −5.36601337338183698204797645110, −4.82589635727517027818009582596, −4.29607697033074311237194309244, −2.92546274460253618429355753390, −2.10098141094549957932904451220, −0.47769402147953363434862378089, 0.47769402147953363434862378089, 2.10098141094549957932904451220, 2.92546274460253618429355753390, 4.29607697033074311237194309244, 4.82589635727517027818009582596, 5.36601337338183698204797645110, 6.04544470125185312587819187384, 6.83204491570017414698661590457, 7.47392474996511657289131602222, 8.093321580282230601036084586032

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