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 − 0.358·3-s + 4-s + 3.12·5-s + 0.358·6-s − 3.38·7-s − 8-s − 2.87·9-s − 3.12·10-s − 3.00·11-s − 0.358·12-s + 5.58·13-s + 3.38·14-s − 1.11·15-s + 16-s + 0.615·17-s + 2.87·18-s − 19-s + 3.12·20-s + 1.21·21-s + 3.00·22-s + 7.17·23-s + 0.358·24-s + 4.76·25-s − 5.58·26-s + 2.10·27-s − 3.38·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.206·3-s + 0.5·4-s + 1.39·5-s + 0.146·6-s − 1.27·7-s − 0.353·8-s − 0.957·9-s − 0.988·10-s − 0.904·11-s − 0.103·12-s + 1.54·13-s + 0.904·14-s − 0.288·15-s + 0.250·16-s + 0.149·17-s + 0.676·18-s − 0.229·19-s + 0.698·20-s + 0.264·21-s + 0.639·22-s + 1.49·23-s + 0.0730·24-s + 0.952·25-s − 1.09·26-s + 0.404·27-s − 0.639·28-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 \( 1 + 0.358T + 3T^{2} \)
5 \( 1 - 3.12T + 5T^{2} \)
7 \( 1 + 3.38T + 7T^{2} \)
11 \( 1 + 3.00T + 11T^{2} \)
13 \( 1 - 5.58T + 13T^{2} \)
17 \( 1 - 0.615T + 17T^{2} \)
23 \( 1 - 7.17T + 23T^{2} \)
29 \( 1 + 1.35T + 29T^{2} \)
31 \( 1 + 6.18T + 31T^{2} \)
37 \( 1 - 4.34T + 37T^{2} \)
41 \( 1 + 4.36T + 41T^{2} \)
43 \( 1 + 1.24T + 43T^{2} \)
47 \( 1 + 6.56T + 47T^{2} \)
53 \( 1 - 2.38T + 53T^{2} \)
59 \( 1 + 0.0783T + 59T^{2} \)
61 \( 1 + 11.8T + 61T^{2} \)
67 \( 1 - 1.93T + 67T^{2} \)
71 \( 1 + 8.93T + 71T^{2} \)
73 \( 1 - 2.99T + 73T^{2} \)
79 \( 1 - 4.20T + 79T^{2} \)
83 \( 1 - 7.46T + 83T^{2} \)
89 \( 1 - 4.41T + 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−7.48629744631030669604110897039, −6.55036397266103565881310398313, −6.21579054519182542639197600878, −5.65703706843918464489991595205, −5.01937369142664922111931468032, −3.48453364237442125154661513116, −3.01461580978705202658647372234, −2.18574344698777699023160671992, −1.17269069168347853067818906263, 0, 1.17269069168347853067818906263, 2.18574344698777699023160671992, 3.01461580978705202658647372234, 3.48453364237442125154661513116, 5.01937369142664922111931468032, 5.65703706843918464489991595205, 6.21579054519182542639197600878, 6.55036397266103565881310398313, 7.48629744631030669604110897039

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