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 + 2.64·3-s + 4-s − 1.93·5-s + 2.64·6-s + 1.07·7-s + 8-s + 3.98·9-s − 1.93·10-s + 1.90·11-s + 2.64·12-s − 2.15·13-s + 1.07·14-s − 5.12·15-s + 16-s + 7.78·17-s + 3.98·18-s − 19-s − 1.93·20-s + 2.82·21-s + 1.90·22-s − 0.222·23-s + 2.64·24-s − 1.24·25-s − 2.15·26-s + 2.60·27-s + 1.07·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.867·5-s + 1.07·6-s + 0.404·7-s + 0.353·8-s + 1.32·9-s − 0.613·10-s + 0.575·11-s + 0.762·12-s − 0.598·13-s + 0.286·14-s − 1.32·15-s + 0.250·16-s + 1.88·17-s + 0.939·18-s − 0.229·19-s − 0.433·20-s + 0.617·21-s + 0.406·22-s − 0.0463·23-s + 0.539·24-s − 0.248·25-s − 0.422·26-s + 0.501·27-s + 0.202·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.802239354$
$L(\frac12)$  $\approx$  $5.802239354$
$L(\frac{3}{2})$   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 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 + 1.93T + 5T^{2} \)
7 \( 1 - 1.07T + 7T^{2} \)
11 \( 1 - 1.90T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 - 7.78T + 17T^{2} \)
23 \( 1 + 0.222T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
31 \( 1 - 8.98T + 31T^{2} \)
37 \( 1 - 4.54T + 37T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 - 2.89T + 43T^{2} \)
47 \( 1 + 1.39T + 47T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 - 1.99T + 59T^{2} \)
61 \( 1 + 3.71T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 5.43T + 71T^{2} \)
73 \( 1 - 5.89T + 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + 0.230T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−7.889248325518905966830903651109, −7.40944127413133836337435405488, −6.58226818484730100637658436003, −5.62645170071654321239777859643, −4.80955905455950789620064131691, −3.98276255951055454429221947601, −3.61694913385100372763863728125, −2.85363001216188285900462641985, −2.09554531354613954340836466654, −1.04618668322711438954940760340, 1.04618668322711438954940760340, 2.09554531354613954340836466654, 2.85363001216188285900462641985, 3.61694913385100372763863728125, 3.98276255951055454429221947601, 4.80955905455950789620064131691, 5.62645170071654321239777859643, 6.58226818484730100637658436003, 7.40944127413133836337435405488, 7.889248325518905966830903651109

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