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 + 0.180·3-s + 4-s + 0.229·5-s + 0.180·6-s − 1.11·7-s + 8-s − 2.96·9-s + 0.229·10-s − 4.26·11-s + 0.180·12-s + 5.52·13-s − 1.11·14-s + 0.0413·15-s + 16-s + 3.05·17-s − 2.96·18-s + 19-s + 0.229·20-s − 0.200·21-s − 4.26·22-s − 0.559·23-s + 0.180·24-s − 4.94·25-s + 5.52·26-s − 1.07·27-s − 1.11·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.104·3-s + 0.5·4-s + 0.102·5-s + 0.0735·6-s − 0.420·7-s + 0.353·8-s − 0.989·9-s + 0.0726·10-s − 1.28·11-s + 0.0520·12-s + 1.53·13-s − 0.297·14-s + 0.0106·15-s + 0.250·16-s + 0.742·17-s − 0.699·18-s + 0.229·19-s + 0.0513·20-s − 0.0437·21-s − 0.910·22-s − 0.116·23-s + 0.0367·24-s − 0.989·25-s + 1.08·26-s − 0.206·27-s − 0.210·28-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 \( 1 - 0.180T + 3T^{2} \)
5 \( 1 - 0.229T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 - 5.52T + 13T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
23 \( 1 + 0.559T + 23T^{2} \)
29 \( 1 + 4.00T + 29T^{2} \)
31 \( 1 - 9.37T + 31T^{2} \)
37 \( 1 - 4.95T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 12.2T + 43T^{2} \)
47 \( 1 + 6.32T + 47T^{2} \)
53 \( 1 + 0.623T + 53T^{2} \)
59 \( 1 - 2.85T + 59T^{2} \)
61 \( 1 + 12.0T + 61T^{2} \)
67 \( 1 + 9.26T + 67T^{2} \)
71 \( 1 - 8.20T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 - 4.31T + 83T^{2} \)
89 \( 1 + 2.30T + 89T^{2} \)
97 \( 1 + 2.67T + 97T^{2} \)
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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

−7.60235683473938237744484290531, −6.34173764900672628078816200413, −6.23130207886353243573450284315, −5.34530419812431020094314947335, −4.86024921638289837347529066542, −3.61109281255887254125696433510, −3.29266479686481312949635336471, −2.48069841055599003491785089902, −1.43508431479748758931349643959, 0, 1.43508431479748758931349643959, 2.48069841055599003491785089902, 3.29266479686481312949635336471, 3.61109281255887254125696433510, 4.86024921638289837347529066542, 5.34530419812431020094314947335, 6.23130207886353243573450284315, 6.34173764900672628078816200413, 7.60235683473938237744484290531

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