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 − 3.08·3-s + 4-s + 2.62·5-s + 3.08·6-s − 3.66·7-s − 8-s + 6.53·9-s − 2.62·10-s + 1.21·11-s − 3.08·12-s − 4.51·13-s + 3.66·14-s − 8.11·15-s + 16-s + 4.39·17-s − 6.53·18-s − 19-s + 2.62·20-s + 11.3·21-s − 1.21·22-s + 2.40·23-s + 3.08·24-s + 1.90·25-s + 4.51·26-s − 10.9·27-s − 3.66·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.78·3-s + 0.5·4-s + 1.17·5-s + 1.26·6-s − 1.38·7-s − 0.353·8-s + 2.17·9-s − 0.831·10-s + 0.367·11-s − 0.891·12-s − 1.25·13-s + 0.979·14-s − 2.09·15-s + 0.250·16-s + 1.06·17-s − 1.54·18-s − 0.229·19-s + 0.587·20-s + 2.47·21-s − 0.259·22-s + 0.500·23-s + 0.630·24-s + 0.381·25-s + 0.885·26-s − 2.10·27-s − 0.692·28-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 \( 1 + 3.08T + 3T^{2} \)
5 \( 1 - 2.62T + 5T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 - 1.21T + 11T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
17 \( 1 - 4.39T + 17T^{2} \)
23 \( 1 - 2.40T + 23T^{2} \)
29 \( 1 + 1.07T + 29T^{2} \)
31 \( 1 + 0.440T + 31T^{2} \)
37 \( 1 - 2.89T + 37T^{2} \)
41 \( 1 + 2.67T + 41T^{2} \)
43 \( 1 + 7.06T + 43T^{2} \)
47 \( 1 + 3.18T + 47T^{2} \)
53 \( 1 - 0.511T + 53T^{2} \)
59 \( 1 - 4.68T + 59T^{2} \)
61 \( 1 - 12.8T + 61T^{2} \)
67 \( 1 - 1.04T + 67T^{2} \)
71 \( 1 + 4.69T + 71T^{2} \)
73 \( 1 - 5.06T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 2.37T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 2.91T + 97T^{2} \)
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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

−7.12547856578107348851519294508, −6.64168591738335590735676740494, −6.29017800254921846336875982413, −5.40142625493302102386722077988, −5.22233294013373333926348632459, −3.96893214190487966978877203901, −2.92409406285726022758285078890, −1.92492384483860458618792624065, −0.941535733235132553151124786780, 0, 0.941535733235132553151124786780, 1.92492384483860458618792624065, 2.92409406285726022758285078890, 3.96893214190487966978877203901, 5.22233294013373333926348632459, 5.40142625493302102386722077988, 6.29017800254921846336875982413, 6.64168591738335590735676740494, 7.12547856578107348851519294508

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