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 + 2.14·3-s + 4-s + 3.16·5-s + 2.14·6-s + 2.76·7-s + 8-s + 1.59·9-s + 3.16·10-s − 4.01·11-s + 2.14·12-s + 5.37·13-s + 2.76·14-s + 6.79·15-s + 16-s + 2.09·17-s + 1.59·18-s − 19-s + 3.16·20-s + 5.93·21-s − 4.01·22-s − 4.42·23-s + 2.14·24-s + 5.04·25-s + 5.37·26-s − 3.00·27-s + 2.76·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.23·3-s + 0.5·4-s + 1.41·5-s + 0.875·6-s + 1.04·7-s + 0.353·8-s + 0.532·9-s + 1.00·10-s − 1.21·11-s + 0.619·12-s + 1.49·13-s + 0.740·14-s + 1.75·15-s + 0.250·16-s + 0.508·17-s + 0.376·18-s − 0.229·19-s + 0.708·20-s + 1.29·21-s − 0.856·22-s − 0.922·23-s + 0.437·24-s + 1.00·25-s + 1.05·26-s − 0.578·27-s + 0.523·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.814900380$
$L(\frac12)$  $\approx$  $7.814900380$
$L(\frac{3}{2})$   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 \( 1 - 2.14T + 3T^{2} \)
5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 - 2.76T + 7T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
13 \( 1 - 5.37T + 13T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
23 \( 1 + 4.42T + 23T^{2} \)
29 \( 1 + 8.75T + 29T^{2} \)
31 \( 1 + 6.73T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + 0.355T + 41T^{2} \)
43 \( 1 - 8.11T + 43T^{2} \)
47 \( 1 - 5.05T + 47T^{2} \)
53 \( 1 - 0.926T + 53T^{2} \)
59 \( 1 - 8.02T + 59T^{2} \)
61 \( 1 - 5.90T + 61T^{2} \)
67 \( 1 - 3.57T + 67T^{2} \)
71 \( 1 + 2.04T + 71T^{2} \)
73 \( 1 - 4.11T + 73T^{2} \)
79 \( 1 + 0.388T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + 13.9T + 89T^{2} \)
97 \( 1 - 2.38T + 97T^{2} \)
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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

−7.86054457997787750377716426212, −7.34955097606568523520841869424, −6.11158621809925168518935617757, −5.71286485609170842883074898234, −5.20627789658811753299623370444, −4.09097053523987678333707765660, −3.54641046137987061788906722130, −2.43161529887406397220148657816, −2.17000198096109227571556797315, −1.31967260134287636779770289313, 1.31967260134287636779770289313, 2.17000198096109227571556797315, 2.43161529887406397220148657816, 3.54641046137987061788906722130, 4.09097053523987678333707765660, 5.20627789658811753299623370444, 5.71286485609170842883074898234, 6.11158621809925168518935617757, 7.34955097606568523520841869424, 7.86054457997787750377716426212

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