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 − 2.41·3-s + 4-s + 0.828·5-s − 2.41·6-s + 0.635·7-s + 8-s + 2.83·9-s + 0.828·10-s + 4.76·11-s − 2.41·12-s + 1.33·13-s + 0.635·14-s − 2.00·15-s + 16-s + 4.13·17-s + 2.83·18-s − 19-s + 0.828·20-s − 1.53·21-s + 4.76·22-s + 8.13·23-s − 2.41·24-s − 4.31·25-s + 1.33·26-s + 0.398·27-s + 0.635·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.39·3-s + 0.5·4-s + 0.370·5-s − 0.986·6-s + 0.240·7-s + 0.353·8-s + 0.945·9-s + 0.261·10-s + 1.43·11-s − 0.697·12-s + 0.371·13-s + 0.169·14-s − 0.516·15-s + 0.250·16-s + 1.00·17-s + 0.668·18-s − 0.229·19-s + 0.185·20-s − 0.334·21-s + 1.01·22-s + 1.69·23-s − 0.493·24-s − 0.862·25-s + 0.262·26-s + 0.0766·27-s + 0.120·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.829704440$
$L(\frac12)$  $\approx$  $2.829704440$
$L(\frac{3}{2})$   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 \( 1 + 2.41T + 3T^{2} \)
5 \( 1 - 0.828T + 5T^{2} \)
7 \( 1 - 0.635T + 7T^{2} \)
11 \( 1 - 4.76T + 11T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 - 4.13T + 17T^{2} \)
23 \( 1 - 8.13T + 23T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 - 6.81T + 31T^{2} \)
37 \( 1 - 0.803T + 37T^{2} \)
41 \( 1 - 9.53T + 41T^{2} \)
43 \( 1 - 0.674T + 43T^{2} \)
47 \( 1 + 3.36T + 47T^{2} \)
53 \( 1 - 6.15T + 53T^{2} \)
59 \( 1 - 6.64T + 59T^{2} \)
61 \( 1 + 1.12T + 61T^{2} \)
67 \( 1 + 1.28T + 67T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 0.475T + 79T^{2} \)
83 \( 1 + 4.38T + 83T^{2} \)
89 \( 1 - 6.45T + 89T^{2} \)
97 \( 1 - 1.33T + 97T^{2} \)
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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

−7.51703210786051623641674521202, −6.82692211273569402095511275736, −6.24953095426284748907806136315, −5.78033255879358170730347539569, −5.15357666156920536334287579599, −4.43028708975778080937283263424, −3.75687282833085226142963662833, −2.78659082462578558498489221146, −1.51678126261319846572492966217, −0.901257334840586146491266477496, 0.901257334840586146491266477496, 1.51678126261319846572492966217, 2.78659082462578558498489221146, 3.75687282833085226142963662833, 4.43028708975778080937283263424, 5.15357666156920536334287579599, 5.78033255879358170730347539569, 6.24953095426284748907806136315, 6.82692211273569402095511275736, 7.51703210786051623641674521202

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