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 − 0.316·3-s + 4-s − 0.633·5-s − 0.316·6-s + 1.30·7-s + 8-s − 2.89·9-s − 0.633·10-s − 3.28·11-s − 0.316·12-s + 2.72·13-s + 1.30·14-s + 0.200·15-s + 16-s + 7.85·17-s − 2.89·18-s − 19-s − 0.633·20-s − 0.412·21-s − 3.28·22-s + 5.94·23-s − 0.316·24-s − 4.59·25-s + 2.72·26-s + 1.86·27-s + 1.30·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.182·3-s + 0.5·4-s − 0.283·5-s − 0.129·6-s + 0.492·7-s + 0.353·8-s − 0.966·9-s − 0.200·10-s − 0.989·11-s − 0.0914·12-s + 0.754·13-s + 0.348·14-s + 0.0518·15-s + 0.250·16-s + 1.90·17-s − 0.683·18-s − 0.229·19-s − 0.141·20-s − 0.0899·21-s − 0.699·22-s + 1.23·23-s − 0.0646·24-s − 0.919·25-s + 0.533·26-s + 0.359·27-s + 0.246·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.844707928$
$L(\frac12)$  $\approx$  $2.844707928$
$L(\frac{3}{2})$   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 \( 1 + 0.316T + 3T^{2} \)
5 \( 1 + 0.633T + 5T^{2} \)
7 \( 1 - 1.30T + 7T^{2} \)
11 \( 1 + 3.28T + 11T^{2} \)
13 \( 1 - 2.72T + 13T^{2} \)
17 \( 1 - 7.85T + 17T^{2} \)
23 \( 1 - 5.94T + 23T^{2} \)
29 \( 1 - 4.86T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 - 6.73T + 41T^{2} \)
43 \( 1 + 8.64T + 43T^{2} \)
47 \( 1 + 4.92T + 47T^{2} \)
53 \( 1 - 6.78T + 53T^{2} \)
59 \( 1 - 6.46T + 59T^{2} \)
61 \( 1 - 8.88T + 61T^{2} \)
67 \( 1 + 9.62T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 9.02T + 83T^{2} \)
89 \( 1 - 1.81T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−7.85735506163881680655266869720, −7.17650187058770157442920287815, −6.22263074204411366454123043316, −5.51223793577246496628435521347, −5.28002904511072303084321595984, −4.33985864780144206482869219291, −3.38415482242283295626470526114, −2.97535191013757664246028066295, −1.88480213409335156353032289792, −0.75814489572299258681022447534, 0.75814489572299258681022447534, 1.88480213409335156353032289792, 2.97535191013757664246028066295, 3.38415482242283295626470526114, 4.33985864780144206482869219291, 5.28002904511072303084321595984, 5.51223793577246496628435521347, 6.22263074204411366454123043316, 7.17650187058770157442920287815, 7.85735506163881680655266869720

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