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 − 2.02·3-s + 4-s + 2.47·5-s − 2.02·6-s + 1.41·7-s + 8-s + 1.11·9-s + 2.47·10-s − 1.90·11-s − 2.02·12-s + 3.04·13-s + 1.41·14-s − 5.01·15-s + 16-s − 0.777·17-s + 1.11·18-s + 19-s + 2.47·20-s − 2.86·21-s − 1.90·22-s − 7.98·23-s − 2.02·24-s + 1.10·25-s + 3.04·26-s + 3.82·27-s + 1.41·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.17·3-s + 0.5·4-s + 1.10·5-s − 0.828·6-s + 0.534·7-s + 0.353·8-s + 0.371·9-s + 0.781·10-s − 0.573·11-s − 0.585·12-s + 0.845·13-s + 0.378·14-s − 1.29·15-s + 0.250·16-s − 0.188·17-s + 0.262·18-s + 0.229·19-s + 0.552·20-s − 0.626·21-s − 0.405·22-s − 1.66·23-s − 0.414·24-s + 0.221·25-s + 0.598·26-s + 0.736·27-s + 0.267·28-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 \( 1 + 2.02T + 3T^{2} \)
5 \( 1 - 2.47T + 5T^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 - 3.04T + 13T^{2} \)
17 \( 1 + 0.777T + 17T^{2} \)
23 \( 1 + 7.98T + 23T^{2} \)
29 \( 1 + 5.89T + 29T^{2} \)
31 \( 1 + 5.52T + 31T^{2} \)
37 \( 1 + 3.09T + 37T^{2} \)
41 \( 1 + 6.61T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 6.04T + 47T^{2} \)
53 \( 1 + 3.46T + 53T^{2} \)
59 \( 1 + 1.73T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 5.11T + 67T^{2} \)
71 \( 1 + 0.661T + 71T^{2} \)
73 \( 1 + 11.5T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + 5.29T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−7.31112703051233282617304194737, −6.36928915990224471048425012942, −5.92995343676606460899839364405, −5.54485309527627481991419042727, −4.91219698774426087049841924582, −4.11071744068249545480833742812, −3.17431455364828796688174420562, −2.04602781415431231535116688762, −1.50942449095690714030855860479, 0, 1.50942449095690714030855860479, 2.04602781415431231535116688762, 3.17431455364828796688174420562, 4.11071744068249545480833742812, 4.91219698774426087049841924582, 5.54485309527627481991419042727, 5.92995343676606460899839364405, 6.36928915990224471048425012942, 7.31112703051233282617304194737

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