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 − 0.219·3-s + 4-s + 2.27·5-s − 0.219·6-s − 3.32·7-s + 8-s − 2.95·9-s + 2.27·10-s + 3.87·11-s − 0.219·12-s − 1.48·13-s − 3.32·14-s − 0.498·15-s + 16-s + 0.776·17-s − 2.95·18-s + 19-s + 2.27·20-s + 0.729·21-s + 3.87·22-s − 8.89·23-s − 0.219·24-s + 0.173·25-s − 1.48·26-s + 1.30·27-s − 3.32·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.126·3-s + 0.5·4-s + 1.01·5-s − 0.0895·6-s − 1.25·7-s + 0.353·8-s − 0.983·9-s + 0.719·10-s + 1.16·11-s − 0.0633·12-s − 0.413·13-s − 0.888·14-s − 0.128·15-s + 0.250·16-s + 0.188·17-s − 0.695·18-s + 0.229·19-s + 0.508·20-s + 0.159·21-s + 0.825·22-s − 1.85·23-s − 0.0447·24-s + 0.0346·25-s − 0.292·26-s + 0.251·27-s − 0.628·28-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 \( 1 + 0.219T + 3T^{2} \)
5 \( 1 - 2.27T + 5T^{2} \)
7 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 - 3.87T + 11T^{2} \)
13 \( 1 + 1.48T + 13T^{2} \)
17 \( 1 - 0.776T + 17T^{2} \)
23 \( 1 + 8.89T + 23T^{2} \)
29 \( 1 - 4.85T + 29T^{2} \)
31 \( 1 - 2.88T + 31T^{2} \)
37 \( 1 + 7.10T + 37T^{2} \)
41 \( 1 + 7.81T + 41T^{2} \)
43 \( 1 - 0.180T + 43T^{2} \)
47 \( 1 - 3.93T + 47T^{2} \)
53 \( 1 - 2.43T + 53T^{2} \)
59 \( 1 + 9.44T + 59T^{2} \)
61 \( 1 + 5.88T + 61T^{2} \)
67 \( 1 + 0.157T + 67T^{2} \)
71 \( 1 - 4.50T + 71T^{2} \)
73 \( 1 + 8.65T + 73T^{2} \)
79 \( 1 + 7.31T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 - 6.08T + 89T^{2} \)
97 \( 1 + 0.915T + 97T^{2} \)
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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

−7.17350935631296732793423703201, −6.47345812312853346955411952331, −6.08377230935166574395951860604, −5.63366410202951661082270028016, −4.71890122157662244501420468585, −3.80058671710974907246775933743, −3.15217956266419736329233295256, −2.40141076840785968200731565365, −1.49426203223721646308967936084, 0, 1.49426203223721646308967936084, 2.40141076840785968200731565365, 3.15217956266419736329233295256, 3.80058671710974907246775933743, 4.71890122157662244501420468585, 5.63366410202951661082270028016, 6.08377230935166574395951860604, 6.47345812312853346955411952331, 7.17350935631296732793423703201

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