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 − 0.785·3-s + 4-s − 1.33·5-s − 0.785·6-s + 4.52·7-s + 8-s − 2.38·9-s − 1.33·10-s + 1.48·11-s − 0.785·12-s − 4.61·13-s + 4.52·14-s + 1.04·15-s + 16-s − 0.964·17-s − 2.38·18-s + 19-s − 1.33·20-s − 3.55·21-s + 1.48·22-s − 0.651·23-s − 0.785·24-s − 3.21·25-s − 4.61·26-s + 4.22·27-s + 4.52·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.453·3-s + 0.5·4-s − 0.596·5-s − 0.320·6-s + 1.71·7-s + 0.353·8-s − 0.794·9-s − 0.422·10-s + 0.448·11-s − 0.226·12-s − 1.27·13-s + 1.20·14-s + 0.270·15-s + 0.250·16-s − 0.234·17-s − 0.561·18-s + 0.229·19-s − 0.298·20-s − 0.775·21-s + 0.317·22-s − 0.135·23-s − 0.160·24-s − 0.643·25-s − 0.905·26-s + 0.813·27-s + 0.855·28-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 \( 1 + 0.785T + 3T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
7 \( 1 - 4.52T + 7T^{2} \)
11 \( 1 - 1.48T + 11T^{2} \)
13 \( 1 + 4.61T + 13T^{2} \)
17 \( 1 + 0.964T + 17T^{2} \)
23 \( 1 + 0.651T + 23T^{2} \)
29 \( 1 + 2.58T + 29T^{2} \)
31 \( 1 + 0.662T + 31T^{2} \)
37 \( 1 - 0.132T + 37T^{2} \)
41 \( 1 - 6.87T + 41T^{2} \)
43 \( 1 + 4.27T + 43T^{2} \)
47 \( 1 - 0.671T + 47T^{2} \)
53 \( 1 + 10.9T + 53T^{2} \)
59 \( 1 - 4.28T + 59T^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 - 8.43T + 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 + 2.16T + 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + 18.0T + 83T^{2} \)
89 \( 1 - 6.72T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−7.56751784545263262172200452255, −6.79439493155034835967808317962, −5.83287257946491087219215400389, −5.35376870879076832740368661178, −4.59854098901144837943941566319, −4.25097783716354066108822903079, −3.15226599502969219222883580014, −2.27748795579752434361539771271, −1.41293242036486651731074345387, 0, 1.41293242036486651731074345387, 2.27748795579752434361539771271, 3.15226599502969219222883580014, 4.25097783716354066108822903079, 4.59854098901144837943941566319, 5.35376870879076832740368661178, 5.83287257946491087219215400389, 6.79439493155034835967808317962, 7.56751784545263262172200452255

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