Properties

Label 2-8018-1.1-c1-0-236
Degree $2$
Conductor $8018$
Sign $1$
Analytic cond. $64.0240$
Root an. cond. $8.00150$
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 + 3.40·3-s + 4-s + 2.94·5-s − 3.40·6-s + 0.502·7-s − 8-s + 8.56·9-s − 2.94·10-s − 2.01·11-s + 3.40·12-s + 4.36·13-s − 0.502·14-s + 10.0·15-s + 16-s + 3.17·17-s − 8.56·18-s + 19-s + 2.94·20-s + 1.70·21-s + 2.01·22-s + 7.38·23-s − 3.40·24-s + 3.67·25-s − 4.36·26-s + 18.9·27-s + 0.502·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.96·3-s + 0.5·4-s + 1.31·5-s − 1.38·6-s + 0.189·7-s − 0.353·8-s + 2.85·9-s − 0.931·10-s − 0.606·11-s + 0.981·12-s + 1.20·13-s − 0.134·14-s + 2.58·15-s + 0.250·16-s + 0.770·17-s − 2.01·18-s + 0.229·19-s + 0.658·20-s + 0.372·21-s + 0.428·22-s + 1.53·23-s − 0.694·24-s + 0.734·25-s − 0.855·26-s + 3.64·27-s + 0.0949·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8018\)    =    \(2 \cdot 19 \cdot 211\)
Sign: $1$
Analytic conductor: \(64.0240\)
Root analytic conductor: \(8.00150\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8018,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.961523291\)
\(L(\frac12)\) \(\approx\) \(4.961523291\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 - T \)
211 \( 1 + T \)
good3 \( 1 - 3.40T + 3T^{2} \)
5 \( 1 - 2.94T + 5T^{2} \)
7 \( 1 - 0.502T + 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 - 4.36T + 13T^{2} \)
17 \( 1 - 3.17T + 17T^{2} \)
23 \( 1 - 7.38T + 23T^{2} \)
29 \( 1 + 3.10T + 29T^{2} \)
31 \( 1 + 9.26T + 31T^{2} \)
37 \( 1 + 0.509T + 37T^{2} \)
41 \( 1 + 7.14T + 41T^{2} \)
43 \( 1 + 2.03T + 43T^{2} \)
47 \( 1 - 7.27T + 47T^{2} \)
53 \( 1 + 2.09T + 53T^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 1.55T + 61T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 6.75T + 73T^{2} \)
79 \( 1 + 3.27T + 79T^{2} \)
83 \( 1 + 8.58T + 83T^{2} \)
89 \( 1 + 7.50T + 89T^{2} \)
97 \( 1 - 3.85T + 97T^{2} \)
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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

−8.062420465396861917266206101564, −7.27384049105197848572223997396, −6.85939864375724368609072872223, −5.78008058795652843074080127277, −5.14273570359517398078888896030, −3.92029231886170090840766768955, −3.19515923185358651136586684984, −2.61905543169216383192369619278, −1.68368804722455377925862020184, −1.33175223643325115253735092747, 1.33175223643325115253735092747, 1.68368804722455377925862020184, 2.61905543169216383192369619278, 3.19515923185358651136586684984, 3.92029231886170090840766768955, 5.14273570359517398078888896030, 5.78008058795652843074080127277, 6.85939864375724368609072872223, 7.27384049105197848572223997396, 8.062420465396861917266206101564

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