Properties

Label 2-8018-1.1-c1-0-232
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.46·3-s + 4-s + 1.82·5-s − 1.46·6-s − 2.77·7-s + 8-s − 0.865·9-s + 1.82·10-s − 1.21·11-s − 1.46·12-s + 1.42·13-s − 2.77·14-s − 2.66·15-s + 16-s + 4.88·17-s − 0.865·18-s + 19-s + 1.82·20-s + 4.05·21-s − 1.21·22-s + 7.05·23-s − 1.46·24-s − 1.67·25-s + 1.42·26-s + 5.64·27-s − 2.77·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.843·3-s + 0.5·4-s + 0.815·5-s − 0.596·6-s − 1.04·7-s + 0.353·8-s − 0.288·9-s + 0.576·10-s − 0.366·11-s − 0.421·12-s + 0.396·13-s − 0.742·14-s − 0.688·15-s + 0.250·16-s + 1.18·17-s − 0.203·18-s + 0.229·19-s + 0.407·20-s + 0.885·21-s − 0.259·22-s + 1.47·23-s − 0.298·24-s − 0.334·25-s + 0.280·26-s + 1.08·27-s − 0.524·28-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: \(1\)
Selberg data: \((2,\ 8018,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.46T + 3T^{2} \)
5 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 + 2.77T + 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 - 1.42T + 13T^{2} \)
17 \( 1 - 4.88T + 17T^{2} \)
23 \( 1 - 7.05T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 + 7.99T + 31T^{2} \)
37 \( 1 + 7.54T + 37T^{2} \)
41 \( 1 + 0.862T + 41T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 - 8.09T + 47T^{2} \)
53 \( 1 - 2.56T + 53T^{2} \)
59 \( 1 + 3.58T + 59T^{2} \)
61 \( 1 + 3.47T + 61T^{2} \)
67 \( 1 - 14.8T + 67T^{2} \)
71 \( 1 + 0.424T + 71T^{2} \)
73 \( 1 - 8.16T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 - 6.13T + 89T^{2} \)
97 \( 1 + 9.03T + 97T^{2} \)
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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

−7.11909062265237472737488745781, −6.67023560548619878006462464527, −5.79950437880825317983857026764, −5.51463856005909147206526861590, −5.08630141859853908719772313158, −3.69678259339430136670911504639, −3.31912383660940084893253005660, −2.34813340585145446113026350942, −1.30359148376382467492414265007, 0, 1.30359148376382467492414265007, 2.34813340585145446113026350942, 3.31912383660940084893253005660, 3.69678259339430136670911504639, 5.08630141859853908719772313158, 5.51463856005909147206526861590, 5.79950437880825317983857026764, 6.67023560548619878006462464527, 7.11909062265237472737488745781

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