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 + 3.11·3-s + 4-s + 3.52·5-s + 3.11·6-s + 1.63·7-s + 8-s + 6.69·9-s + 3.52·10-s − 1.92·11-s + 3.11·12-s − 4.26·13-s + 1.63·14-s + 10.9·15-s + 16-s − 0.309·17-s + 6.69·18-s − 19-s + 3.52·20-s + 5.08·21-s − 1.92·22-s + 7.57·23-s + 3.11·24-s + 7.45·25-s − 4.26·26-s + 11.4·27-s + 1.63·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.79·3-s + 0.5·4-s + 1.57·5-s + 1.27·6-s + 0.617·7-s + 0.353·8-s + 2.23·9-s + 1.11·10-s − 0.580·11-s + 0.898·12-s − 1.18·13-s + 0.436·14-s + 2.83·15-s + 0.250·16-s − 0.0749·17-s + 1.57·18-s − 0.229·19-s + 0.789·20-s + 1.10·21-s − 0.410·22-s + 1.57·23-s + 0.635·24-s + 1.49·25-s − 0.836·26-s + 2.21·27-s + 0.308·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.260499038$
$L(\frac12)$  $\approx$  $9.260499038$
$L(\frac{3}{2})$   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 \( 1 - 3.11T + 3T^{2} \)
5 \( 1 - 3.52T + 5T^{2} \)
7 \( 1 - 1.63T + 7T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + 4.26T + 13T^{2} \)
17 \( 1 + 0.309T + 17T^{2} \)
23 \( 1 - 7.57T + 23T^{2} \)
29 \( 1 + 8.60T + 29T^{2} \)
31 \( 1 - 0.310T + 31T^{2} \)
37 \( 1 + 2.33T + 37T^{2} \)
41 \( 1 - 0.172T + 41T^{2} \)
43 \( 1 - 5.61T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 13.4T + 61T^{2} \)
67 \( 1 + 4.34T + 67T^{2} \)
71 \( 1 + 5.01T + 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 - 1.05T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 0.906T + 97T^{2} \)
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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

−7.63639841211409435195448080012, −7.32834634389329739712748358181, −6.54131436849143635499039400907, −5.50225197601856877120868494279, −5.02680011252260892775833424195, −4.28940225497498697870896543373, −3.24563223675261380105122638950, −2.62876519180677309435629660644, −2.09365951667412626839075917614, −1.46672622725348377443502697480, 1.46672622725348377443502697480, 2.09365951667412626839075917614, 2.62876519180677309435629660644, 3.24563223675261380105122638950, 4.28940225497498697870896543373, 5.02680011252260892775833424195, 5.50225197601856877120868494279, 6.54131436849143635499039400907, 7.32834634389329739712748358181, 7.63639841211409435195448080012

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