Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 1.37·3-s + 4-s + 3.53·5-s + 1.37·6-s − 2.08·7-s + 8-s − 1.10·9-s + 3.53·10-s − 3.14·11-s + 1.37·12-s − 3.99·13-s − 2.08·14-s + 4.87·15-s + 16-s − 6.01·17-s − 1.10·18-s + 19-s + 3.53·20-s − 2.87·21-s − 3.14·22-s − 6.22·23-s + 1.37·24-s + 7.51·25-s − 3.99·26-s − 5.65·27-s − 2.08·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.795·3-s + 0.5·4-s + 1.58·5-s + 0.562·6-s − 0.788·7-s + 0.353·8-s − 0.367·9-s + 1.11·10-s − 0.947·11-s + 0.397·12-s − 1.10·13-s − 0.557·14-s + 1.25·15-s + 0.250·16-s − 1.45·17-s − 0.259·18-s + 0.229·19-s + 0.790·20-s − 0.626·21-s − 0.670·22-s − 1.29·23-s + 0.281·24-s + 1.50·25-s − 0.783·26-s − 1.08·27-s − 0.394·28-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  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 1.37T + 3T^{2} \)
5 \( 1 - 3.53T + 5T^{2} \)
7 \( 1 + 2.08T + 7T^{2} \)
11 \( 1 + 3.14T + 11T^{2} \)
13 \( 1 + 3.99T + 13T^{2} \)
17 \( 1 + 6.01T + 17T^{2} \)
23 \( 1 + 6.22T + 23T^{2} \)
29 \( 1 + 6.86T + 29T^{2} \)
31 \( 1 - 6.59T + 31T^{2} \)
37 \( 1 + 3.67T + 37T^{2} \)
41 \( 1 - 8.95T + 41T^{2} \)
43 \( 1 - 0.304T + 43T^{2} \)
47 \( 1 - 0.829T + 47T^{2} \)
53 \( 1 - 3.90T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 2.06T + 61T^{2} \)
67 \( 1 - 4.91T + 67T^{2} \)
71 \( 1 + 1.61T + 71T^{2} \)
73 \( 1 + 2.42T + 73T^{2} \)
79 \( 1 + 2.78T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + 3.09T + 89T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−7.44641789843387423648999679136, −6.58618155687846105968623307483, −6.03739356320367427493141288541, −5.46819933396954558633108704246, −4.75197346371912589098075668575, −3.82356804009347441759591375191, −2.77853141659500952639448813204, −2.47173287111427554462641741424, −1.88034744274486959387721623832, 0, 1.88034744274486959387721623832, 2.47173287111427554462641741424, 2.77853141659500952639448813204, 3.82356804009347441759591375191, 4.75197346371912589098075668575, 5.46819933396954558633108704246, 6.03739356320367427493141288541, 6.58618155687846105968623307483, 7.44641789843387423648999679136

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