Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.40·3-s + 4-s + 3.63·5-s − 1.40·6-s − 3.46·7-s − 8-s − 1.03·9-s − 3.63·10-s + 4.21·11-s + 1.40·12-s − 4.66·13-s + 3.46·14-s + 5.10·15-s + 16-s + 1.56·17-s + 1.03·18-s − 19-s + 3.63·20-s − 4.85·21-s − 4.21·22-s − 1.74·23-s − 1.40·24-s + 8.21·25-s + 4.66·26-s − 5.65·27-s − 3.46·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.810·3-s + 0.5·4-s + 1.62·5-s − 0.572·6-s − 1.30·7-s − 0.353·8-s − 0.343·9-s − 1.14·10-s + 1.27·11-s + 0.405·12-s − 1.29·13-s + 0.924·14-s + 1.31·15-s + 0.250·16-s + 0.379·17-s + 0.243·18-s − 0.229·19-s + 0.812·20-s − 1.05·21-s − 0.899·22-s − 0.364·23-s − 0.286·24-s + 1.64·25-s + 0.914·26-s − 1.08·27-s − 0.653·28-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

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
19 \( 1 + T \)
211 \( 1 + T \)
good3 \( 1 - 1.40T + 3T^{2} \)
5 \( 1 - 3.63T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 4.21T + 11T^{2} \)
13 \( 1 + 4.66T + 13T^{2} \)
17 \( 1 - 1.56T + 17T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 - 3.59T + 29T^{2} \)
31 \( 1 + 7.86T + 31T^{2} \)
37 \( 1 - 1.13T + 37T^{2} \)
41 \( 1 - 1.01T + 41T^{2} \)
43 \( 1 + 6.56T + 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 + 12.4T + 53T^{2} \)
59 \( 1 + 6.49T + 59T^{2} \)
61 \( 1 + 0.292T + 61T^{2} \)
67 \( 1 + 13.5T + 67T^{2} \)
71 \( 1 + 2.17T + 71T^{2} \)
73 \( 1 - 5.79T + 73T^{2} \)
79 \( 1 + 5.89T + 79T^{2} \)
83 \( 1 + 6.45T + 83T^{2} \)
89 \( 1 + 6.34T + 89T^{2} \)
97 \( 1 - 0.532T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.42777807954507513294328632634, −6.85361005394718955324734713343, −6.11042825668094772574402979508, −5.81205059096756496079861554691, −4.70253701051658185330374722195, −3.48778829022516434707518546089, −2.92794127212382638060221186988, −2.18719027035832986546122540385, −1.48576691762480650451437680317, 0, 1.48576691762480650451437680317, 2.18719027035832986546122540385, 2.92794127212382638060221186988, 3.48778829022516434707518546089, 4.70253701051658185330374722195, 5.81205059096756496079861554691, 6.11042825668094772574402979508, 6.85361005394718955324734713343, 7.42777807954507513294328632634

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