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 + 0.625·3-s + 4-s − 0.793·5-s + 0.625·6-s − 2.19·7-s + 8-s − 2.60·9-s − 0.793·10-s + 1.89·11-s + 0.625·12-s + 7.04·13-s − 2.19·14-s − 0.496·15-s + 16-s + 7.96·17-s − 2.60·18-s − 19-s − 0.793·20-s − 1.37·21-s + 1.89·22-s + 1.50·23-s + 0.625·24-s − 4.37·25-s + 7.04·26-s − 3.50·27-s − 2.19·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.361·3-s + 0.5·4-s − 0.354·5-s + 0.255·6-s − 0.829·7-s + 0.353·8-s − 0.869·9-s − 0.250·10-s + 0.570·11-s + 0.180·12-s + 1.95·13-s − 0.586·14-s − 0.128·15-s + 0.250·16-s + 1.93·17-s − 0.614·18-s − 0.229·19-s − 0.177·20-s − 0.299·21-s + 0.403·22-s + 0.314·23-s + 0.127·24-s − 0.874·25-s + 1.38·26-s − 0.675·27-s − 0.414·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.461227469$
$L(\frac12)$  $\approx$  $3.461227469$
$L(\frac{3}{2})$   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 \( 1 - 0.625T + 3T^{2} \)
5 \( 1 + 0.793T + 5T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
11 \( 1 - 1.89T + 11T^{2} \)
13 \( 1 - 7.04T + 13T^{2} \)
17 \( 1 - 7.96T + 17T^{2} \)
23 \( 1 - 1.50T + 23T^{2} \)
29 \( 1 + 3.88T + 29T^{2} \)
31 \( 1 - 1.39T + 31T^{2} \)
37 \( 1 + 5.93T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 3.18T + 53T^{2} \)
59 \( 1 - 3.36T + 59T^{2} \)
61 \( 1 - 5.10T + 61T^{2} \)
67 \( 1 - 3.86T + 67T^{2} \)
71 \( 1 - 5.10T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 3.71T + 83T^{2} \)
89 \( 1 + 2.03T + 89T^{2} \)
97 \( 1 - 15.5T + 97T^{2} \)
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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

−7.78333647631646844402511583358, −7.09820891273657583996297815587, −6.16157782597415491023750438990, −5.90069675519680186458644428411, −5.16574917314931732240619924494, −3.78065037787998933836566097494, −3.67181125349319172097914328257, −3.07077015333357996717586987227, −1.89797304772003857429221914576, −0.826155290570608958576189721809, 0.826155290570608958576189721809, 1.89797304772003857429221914576, 3.07077015333357996717586987227, 3.67181125349319172097914328257, 3.78065037787998933836566097494, 5.16574917314931732240619924494, 5.90069675519680186458644428411, 6.16157782597415491023750438990, 7.09820891273657583996297815587, 7.78333647631646844402511583358

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