Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3.30·3-s + 4-s − 3.07·5-s + 3.30·6-s − 4.11·7-s + 8-s + 7.89·9-s − 3.07·10-s − 4.16·11-s + 3.30·12-s − 6.00·13-s − 4.11·14-s − 10.1·15-s + 16-s + 4.19·17-s + 7.89·18-s − 19-s − 3.07·20-s − 13.5·21-s − 4.16·22-s + 7.28·23-s + 3.30·24-s + 4.47·25-s − 6.00·26-s + 16.1·27-s − 4.11·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.90·3-s + 0.5·4-s − 1.37·5-s + 1.34·6-s − 1.55·7-s + 0.353·8-s + 2.63·9-s − 0.973·10-s − 1.25·11-s + 0.952·12-s − 1.66·13-s − 1.09·14-s − 2.62·15-s + 0.250·16-s + 1.01·17-s + 1.86·18-s − 0.229·19-s − 0.688·20-s − 2.96·21-s − 0.888·22-s + 1.51·23-s + 0.673·24-s + 0.895·25-s − 1.17·26-s + 3.10·27-s − 0.776·28-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  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.905602116$
$L(\frac12)$  $\approx$  $3.905602116$
$L(\frac{3}{2})$   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 \( 1 - 3.30T + 3T^{2} \)
5 \( 1 + 3.07T + 5T^{2} \)
7 \( 1 + 4.11T + 7T^{2} \)
11 \( 1 + 4.16T + 11T^{2} \)
13 \( 1 + 6.00T + 13T^{2} \)
17 \( 1 - 4.19T + 17T^{2} \)
23 \( 1 - 7.28T + 23T^{2} \)
29 \( 1 - 9.67T + 29T^{2} \)
31 \( 1 + 2.11T + 31T^{2} \)
37 \( 1 - 8.60T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 - 6.48T + 43T^{2} \)
47 \( 1 - 8.80T + 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 5.32T + 61T^{2} \)
67 \( 1 - 2.92T + 67T^{2} \)
71 \( 1 + 1.20T + 71T^{2} \)
73 \( 1 + 9.70T + 73T^{2} \)
79 \( 1 + 8.78T + 79T^{2} \)
83 \( 1 - 2.66T + 83T^{2} \)
89 \( 1 + 5.01T + 89T^{2} \)
97 \( 1 + 7.76T + 97T^{2} \)
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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

−7.77979110763607978201112838746, −7.25406787440519511552139517174, −6.88248358283947036006233228267, −5.65283821388291352427094534186, −4.56266153419286481808722147163, −4.21998180179090896836618566873, −3.24854305146077828315408511085, −2.77417440498180207615663927005, −2.61457500574900883160080148402, −0.76665021211034476944058435062, 0.76665021211034476944058435062, 2.61457500574900883160080148402, 2.77417440498180207615663927005, 3.24854305146077828315408511085, 4.21998180179090896836618566873, 4.56266153419286481808722147163, 5.65283821388291352427094534186, 6.88248358283947036006233228267, 7.25406787440519511552139517174, 7.77979110763607978201112838746

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