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 − 1.14·3-s + 4-s + 2.89·5-s − 1.14·6-s + 3.60·7-s + 8-s − 1.68·9-s + 2.89·10-s + 3.08·11-s − 1.14·12-s + 1.07·13-s + 3.60·14-s − 3.32·15-s + 16-s + 2.58·17-s − 1.68·18-s − 19-s + 2.89·20-s − 4.13·21-s + 3.08·22-s − 1.16·23-s − 1.14·24-s + 3.36·25-s + 1.07·26-s + 5.37·27-s + 3.60·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.662·3-s + 0.5·4-s + 1.29·5-s − 0.468·6-s + 1.36·7-s + 0.353·8-s − 0.560·9-s + 0.914·10-s + 0.930·11-s − 0.331·12-s + 0.297·13-s + 0.963·14-s − 0.857·15-s + 0.250·16-s + 0.626·17-s − 0.396·18-s − 0.229·19-s + 0.646·20-s − 0.903·21-s + 0.657·22-s − 0.242·23-s − 0.234·24-s + 0.673·25-s + 0.210·26-s + 1.03·27-s + 0.681·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.467320906$
$L(\frac12)$  $\approx$  $4.467320906$
$L(\frac{3}{2})$   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 \( 1 + 1.14T + 3T^{2} \)
5 \( 1 - 2.89T + 5T^{2} \)
7 \( 1 - 3.60T + 7T^{2} \)
11 \( 1 - 3.08T + 11T^{2} \)
13 \( 1 - 1.07T + 13T^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 5.13T + 29T^{2} \)
31 \( 1 + 0.677T + 31T^{2} \)
37 \( 1 + 2.27T + 37T^{2} \)
41 \( 1 + 3.80T + 41T^{2} \)
43 \( 1 - 1.52T + 43T^{2} \)
47 \( 1 - 1.09T + 47T^{2} \)
53 \( 1 - 11.0T + 53T^{2} \)
59 \( 1 - 7.21T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 8.91T + 71T^{2} \)
73 \( 1 - 3.33T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 4.72T + 83T^{2} \)
89 \( 1 + 0.242T + 89T^{2} \)
97 \( 1 - 0.265T + 97T^{2} \)
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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

−7.71738723261984273888068465402, −6.87618715501485424882716241311, −6.11075588354248501832079542119, −5.81185729002915916788536050145, −5.08898764111127411184604828245, −4.57893161150547478981847007041, −3.61104695435015851782348601511, −2.57934108324374437138626236891, −1.76007926848280292273210703853, −1.07030158927293895282753892596, 1.07030158927293895282753892596, 1.76007926848280292273210703853, 2.57934108324374437138626236891, 3.61104695435015851782348601511, 4.57893161150547478981847007041, 5.08898764111127411184604828245, 5.81185729002915916788536050145, 6.11075588354248501832079542119, 6.87618715501485424882716241311, 7.71738723261984273888068465402

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