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 + 2.76·3-s + 4-s − 1.78·5-s + 2.76·6-s − 4.04·7-s + 8-s + 4.64·9-s − 1.78·10-s − 1.77·11-s + 2.76·12-s + 2.72·13-s − 4.04·14-s − 4.93·15-s + 16-s − 2.11·17-s + 4.64·18-s + 19-s − 1.78·20-s − 11.1·21-s − 1.77·22-s + 3.01·23-s + 2.76·24-s − 1.81·25-s + 2.72·26-s + 4.56·27-s − 4.04·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.59·3-s + 0.5·4-s − 0.797·5-s + 1.12·6-s − 1.52·7-s + 0.353·8-s + 1.54·9-s − 0.564·10-s − 0.534·11-s + 0.798·12-s + 0.757·13-s − 1.08·14-s − 1.27·15-s + 0.250·16-s − 0.512·17-s + 1.09·18-s + 0.229·19-s − 0.398·20-s − 2.44·21-s − 0.378·22-s + 0.628·23-s + 0.564·24-s − 0.363·25-s + 0.535·26-s + 0.877·27-s − 0.764·28-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 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 + 1.78T + 5T^{2} \)
7 \( 1 + 4.04T + 7T^{2} \)
11 \( 1 + 1.77T + 11T^{2} \)
13 \( 1 - 2.72T + 13T^{2} \)
17 \( 1 + 2.11T + 17T^{2} \)
23 \( 1 - 3.01T + 23T^{2} \)
29 \( 1 + 6.86T + 29T^{2} \)
31 \( 1 + 4.16T + 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 0.0551T + 43T^{2} \)
47 \( 1 - 8.03T + 47T^{2} \)
53 \( 1 + 1.47T + 53T^{2} \)
59 \( 1 + 11.6T + 59T^{2} \)
61 \( 1 + 4.45T + 61T^{2} \)
67 \( 1 - 0.780T + 67T^{2} \)
71 \( 1 - 0.868T + 71T^{2} \)
73 \( 1 - 5.79T + 73T^{2} \)
79 \( 1 + 8.77T + 79T^{2} \)
83 \( 1 + 4.95T + 83T^{2} \)
89 \( 1 + 5.44T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−7.45099823226925950744074265691, −6.97869875966928667174477566775, −6.17244258923802144958602506143, −5.36492906280660633736164127065, −4.21150265664622431610191891131, −3.74849517274626114579644015414, −3.19194598856328121486750339874, −2.67290641435814870264133151292, −1.65104188744116694953331636561, 0, 1.65104188744116694953331636561, 2.67290641435814870264133151292, 3.19194598856328121486750339874, 3.74849517274626114579644015414, 4.21150265664622431610191891131, 5.36492906280660633736164127065, 6.17244258923802144958602506143, 6.97869875966928667174477566775, 7.45099823226925950744074265691

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