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 − 0.833·3-s + 4-s − 3.18·5-s − 0.833·6-s + 0.455·7-s + 8-s − 2.30·9-s − 3.18·10-s − 1.37·11-s − 0.833·12-s + 5.32·13-s + 0.455·14-s + 2.65·15-s + 16-s − 7.33·17-s − 2.30·18-s − 19-s − 3.18·20-s − 0.379·21-s − 1.37·22-s − 4.82·23-s − 0.833·24-s + 5.14·25-s + 5.32·26-s + 4.42·27-s + 0.455·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.481·3-s + 0.5·4-s − 1.42·5-s − 0.340·6-s + 0.171·7-s + 0.353·8-s − 0.768·9-s − 1.00·10-s − 0.414·11-s − 0.240·12-s + 1.47·13-s + 0.121·14-s + 0.685·15-s + 0.250·16-s − 1.77·17-s − 0.543·18-s − 0.229·19-s − 0.712·20-s − 0.0827·21-s − 0.292·22-s − 1.00·23-s − 0.170·24-s + 1.02·25-s + 1.04·26-s + 0.850·27-s + 0.0859·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.157403734$
$L(\frac12)$  $\approx$  $1.157403734$
$L(\frac{3}{2})$   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 \( 1 + 0.833T + 3T^{2} \)
5 \( 1 + 3.18T + 5T^{2} \)
7 \( 1 - 0.455T + 7T^{2} \)
11 \( 1 + 1.37T + 11T^{2} \)
13 \( 1 - 5.32T + 13T^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
23 \( 1 + 4.82T + 23T^{2} \)
29 \( 1 - 1.65T + 29T^{2} \)
31 \( 1 - 5.97T + 31T^{2} \)
37 \( 1 + 8.53T + 37T^{2} \)
41 \( 1 + 2.32T + 41T^{2} \)
43 \( 1 + 5.06T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 - 0.902T + 53T^{2} \)
59 \( 1 + 0.573T + 59T^{2} \)
61 \( 1 + 0.904T + 61T^{2} \)
67 \( 1 - 7.39T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 1.44T + 73T^{2} \)
79 \( 1 + 1.09T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 + 7.72T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−7.930158187637153543974785588629, −6.85360969732061361828552602912, −6.49598880214706610857520276116, −5.73042139531229665810848518153, −4.90597640975598665059131233835, −4.28916693424080635160513558422, −3.67248379671760689597156289529, −2.93088205226353662162299934063, −1.87996810239904176589594306626, −0.47105753643737937901661713869, 0.47105753643737937901661713869, 1.87996810239904176589594306626, 2.93088205226353662162299934063, 3.67248379671760689597156289529, 4.28916693424080635160513558422, 4.90597640975598665059131233835, 5.73042139531229665810848518153, 6.49598880214706610857520276116, 6.85360969732061361828552602912, 7.930158187637153543974785588629

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