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 − 1.51·3-s + 4-s − 2.91·5-s − 1.51·6-s − 3.75·7-s + 8-s − 0.707·9-s − 2.91·10-s − 1.07·11-s − 1.51·12-s + 3.84·13-s − 3.75·14-s + 4.40·15-s + 16-s + 1.46·17-s − 0.707·18-s + 19-s − 2.91·20-s + 5.67·21-s − 1.07·22-s − 1.57·23-s − 1.51·24-s + 3.47·25-s + 3.84·26-s + 5.61·27-s − 3.75·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.874·3-s + 0.5·4-s − 1.30·5-s − 0.618·6-s − 1.41·7-s + 0.353·8-s − 0.235·9-s − 0.920·10-s − 0.324·11-s − 0.437·12-s + 1.06·13-s − 1.00·14-s + 1.13·15-s + 0.250·16-s + 0.354·17-s − 0.166·18-s + 0.229·19-s − 0.651·20-s + 1.23·21-s − 0.229·22-s − 0.327·23-s − 0.309·24-s + 0.695·25-s + 0.754·26-s + 1.08·27-s − 0.708·28-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 \( 1 + 1.51T + 3T^{2} \)
5 \( 1 + 2.91T + 5T^{2} \)
7 \( 1 + 3.75T + 7T^{2} \)
11 \( 1 + 1.07T + 11T^{2} \)
13 \( 1 - 3.84T + 13T^{2} \)
17 \( 1 - 1.46T + 17T^{2} \)
23 \( 1 + 1.57T + 23T^{2} \)
29 \( 1 - 2.48T + 29T^{2} \)
31 \( 1 + 5.80T + 31T^{2} \)
37 \( 1 - 8.92T + 37T^{2} \)
41 \( 1 + 3.70T + 41T^{2} \)
43 \( 1 - 2.26T + 43T^{2} \)
47 \( 1 - 3.11T + 47T^{2} \)
53 \( 1 + 4.03T + 53T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 - 6.86T + 61T^{2} \)
67 \( 1 - 7.13T + 67T^{2} \)
71 \( 1 - 4.01T + 71T^{2} \)
73 \( 1 + 15.5T + 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 0.410T + 83T^{2} \)
89 \( 1 + 1.42T + 89T^{2} \)
97 \( 1 + 7.18T + 97T^{2} \)
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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

−7.20988233072495587054386361854, −6.72465152441924712798139816676, −5.90756053300504111771086000592, −5.62858649900914324653095195601, −4.59720137136577457654674870112, −3.83185100224069239857193395595, −3.37716482177513432630213142512, −2.58901127714141792470147042054, −0.946147187977739536330686042913, 0, 0.946147187977739536330686042913, 2.58901127714141792470147042054, 3.37716482177513432630213142512, 3.83185100224069239857193395595, 4.59720137136577457654674870112, 5.62858649900914324653095195601, 5.90756053300504111771086000592, 6.72465152441924712798139816676, 7.20988233072495587054386361854

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