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 − 2.56·3-s + 4-s − 1.11·5-s − 2.56·6-s − 1.44·7-s + 8-s + 3.59·9-s − 1.11·10-s − 1.59·11-s − 2.56·12-s + 3.12·13-s − 1.44·14-s + 2.87·15-s + 16-s − 5.27·17-s + 3.59·18-s + 19-s − 1.11·20-s + 3.72·21-s − 1.59·22-s − 0.783·23-s − 2.56·24-s − 3.74·25-s + 3.12·26-s − 1.53·27-s − 1.44·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.48·3-s + 0.5·4-s − 0.500·5-s − 1.04·6-s − 0.547·7-s + 0.353·8-s + 1.19·9-s − 0.354·10-s − 0.481·11-s − 0.741·12-s + 0.867·13-s − 0.387·14-s + 0.742·15-s + 0.250·16-s − 1.27·17-s + 0.848·18-s + 0.229·19-s − 0.250·20-s + 0.812·21-s − 0.340·22-s − 0.163·23-s − 0.524·24-s − 0.749·25-s + 0.613·26-s − 0.295·27-s − 0.273·28-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 \( 1 + 2.56T + 3T^{2} \)
5 \( 1 + 1.11T + 5T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + 1.59T + 11T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 + 5.27T + 17T^{2} \)
23 \( 1 + 0.783T + 23T^{2} \)
29 \( 1 - 8.48T + 29T^{2} \)
31 \( 1 - 4.24T + 31T^{2} \)
37 \( 1 + 7.23T + 37T^{2} \)
41 \( 1 + 0.308T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 - 1.70T + 47T^{2} \)
53 \( 1 - 6.97T + 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + 0.529T + 61T^{2} \)
67 \( 1 - 4.61T + 67T^{2} \)
71 \( 1 + 2.79T + 71T^{2} \)
73 \( 1 - 13.8T + 73T^{2} \)
79 \( 1 + 5.27T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 + 4.76T + 89T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−7.14710942516158362846946808453, −6.51049215378898468694644356171, −6.14069104269770513728918359562, −5.43023002896911015281573850738, −4.68478571733060601843181953469, −4.16155452657975320290564775854, −3.29348224783181520012726336393, −2.32825311895760005036357475889, −1.03808353825987832176935234206, 0, 1.03808353825987832176935234206, 2.32825311895760005036357475889, 3.29348224783181520012726336393, 4.16155452657975320290564775854, 4.68478571733060601843181953469, 5.43023002896911015281573850738, 6.14069104269770513728918359562, 6.51049215378898468694644356171, 7.14710942516158362846946808453

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