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 − 2.00·3-s + 4-s − 1.79·5-s − 2.00·6-s − 2.43·7-s + 8-s + 1.03·9-s − 1.79·10-s − 1.56·11-s − 2.00·12-s − 4.34·13-s − 2.43·14-s + 3.60·15-s + 16-s − 2.98·17-s + 1.03·18-s − 19-s − 1.79·20-s + 4.89·21-s − 1.56·22-s + 1.92·23-s − 2.00·24-s − 1.78·25-s − 4.34·26-s + 3.95·27-s − 2.43·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.802·5-s − 0.819·6-s − 0.921·7-s + 0.353·8-s + 0.343·9-s − 0.567·10-s − 0.472·11-s − 0.579·12-s − 1.20·13-s − 0.651·14-s + 0.929·15-s + 0.250·16-s − 0.724·17-s + 0.242·18-s − 0.229·19-s − 0.401·20-s + 1.06·21-s − 0.334·22-s + 0.400·23-s − 0.409·24-s − 0.356·25-s − 0.851·26-s + 0.760·27-s − 0.460·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1362080857$
$L(\frac12)$  $\approx$  $0.1362080857$
$L(\frac{3}{2})$   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 \( 1 + 2.00T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 + 1.56T + 11T^{2} \)
13 \( 1 + 4.34T + 13T^{2} \)
17 \( 1 + 2.98T + 17T^{2} \)
23 \( 1 - 1.92T + 23T^{2} \)
29 \( 1 - 0.562T + 29T^{2} \)
31 \( 1 + 4.85T + 31T^{2} \)
37 \( 1 + 3.93T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 8.43T + 43T^{2} \)
47 \( 1 + 6.03T + 47T^{2} \)
53 \( 1 + 4.46T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 + 12.9T + 61T^{2} \)
67 \( 1 + 15.9T + 67T^{2} \)
71 \( 1 - 8.05T + 71T^{2} \)
73 \( 1 - 2.43T + 73T^{2} \)
79 \( 1 + 2.99T + 79T^{2} \)
83 \( 1 - 9.69T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 - 3.44T + 97T^{2} \)
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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

−7.58165345197582289914601848416, −6.77847613740987211125871291954, −6.58326284974081863155553367799, −5.61132650021232063664217848759, −5.03489600002570827468250383593, −4.52230491609037296156401715074, −3.53625979850250200573127002415, −2.93885785142970422991897240146, −1.84793415044383026942513430377, −0.16246472425579553278270564490, 0.16246472425579553278270564490, 1.84793415044383026942513430377, 2.93885785142970422991897240146, 3.53625979850250200573127002415, 4.52230491609037296156401715074, 5.03489600002570827468250383593, 5.61132650021232063664217848759, 6.58326284974081863155553367799, 6.77847613740987211125871291954, 7.58165345197582289914601848416

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