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 − 1.44·3-s + 4-s − 1.29·5-s − 1.44·6-s + 1.46·7-s + 8-s − 0.920·9-s − 1.29·10-s + 4.03·11-s − 1.44·12-s − 4.55·13-s + 1.46·14-s + 1.86·15-s + 16-s + 4.46·17-s − 0.920·18-s + 19-s − 1.29·20-s − 2.11·21-s + 4.03·22-s − 5.71·23-s − 1.44·24-s − 3.32·25-s − 4.55·26-s + 5.65·27-s + 1.46·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.832·3-s + 0.5·4-s − 0.579·5-s − 0.588·6-s + 0.554·7-s + 0.353·8-s − 0.306·9-s − 0.409·10-s + 1.21·11-s − 0.416·12-s − 1.26·13-s + 0.391·14-s + 0.482·15-s + 0.250·16-s + 1.08·17-s − 0.216·18-s + 0.229·19-s − 0.289·20-s − 0.461·21-s + 0.859·22-s − 1.19·23-s − 0.294·24-s − 0.664·25-s − 0.893·26-s + 1.08·27-s + 0.277·28-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 \( 1 + 1.44T + 3T^{2} \)
5 \( 1 + 1.29T + 5T^{2} \)
7 \( 1 - 1.46T + 7T^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + 4.55T + 13T^{2} \)
17 \( 1 - 4.46T + 17T^{2} \)
23 \( 1 + 5.71T + 23T^{2} \)
29 \( 1 + 2.18T + 29T^{2} \)
31 \( 1 - 2.09T + 31T^{2} \)
37 \( 1 - 0.932T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 5.26T + 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 - 6.23T + 59T^{2} \)
61 \( 1 - 7.92T + 61T^{2} \)
67 \( 1 + 5.85T + 67T^{2} \)
71 \( 1 + 9.87T + 71T^{2} \)
73 \( 1 + 1.85T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + 6.53T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−7.34670051419170451539759772189, −6.70139096563388124732973077184, −5.94977794863308233811761928836, −5.36928954345595440365629042327, −4.74958430294467162227086738873, −4.00313341387436601686340669292, −3.34405380535419126476094014765, −2.26704506980568438905967288037, −1.26706232446887645713970544660, 0, 1.26706232446887645713970544660, 2.26704506980568438905967288037, 3.34405380535419126476094014765, 4.00313341387436601686340669292, 4.74958430294467162227086738873, 5.36928954345595440365629042327, 5.94977794863308233811761928836, 6.70139096563388124732973077184, 7.34670051419170451539759772189

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