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 − 1.14·3-s + 4-s + 1.90·5-s − 1.14·6-s + 2.39·7-s + 8-s − 1.69·9-s + 1.90·10-s − 4.82·11-s − 1.14·12-s − 2.93·13-s + 2.39·14-s − 2.16·15-s + 16-s − 2.32·17-s − 1.69·18-s + 19-s + 1.90·20-s − 2.73·21-s − 4.82·22-s + 3.20·23-s − 1.14·24-s − 1.38·25-s − 2.93·26-s + 5.35·27-s + 2.39·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.658·3-s + 0.5·4-s + 0.850·5-s − 0.465·6-s + 0.904·7-s + 0.353·8-s − 0.566·9-s + 0.601·10-s − 1.45·11-s − 0.329·12-s − 0.815·13-s + 0.639·14-s − 0.559·15-s + 0.250·16-s − 0.563·17-s − 0.400·18-s + 0.229·19-s + 0.425·20-s − 0.595·21-s − 1.02·22-s + 0.667·23-s − 0.232·24-s − 0.276·25-s − 0.576·26-s + 1.03·27-s + 0.452·28-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 \( 1 + 1.14T + 3T^{2} \)
5 \( 1 - 1.90T + 5T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 + 4.82T + 11T^{2} \)
13 \( 1 + 2.93T + 13T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
23 \( 1 - 3.20T + 23T^{2} \)
29 \( 1 + 0.604T + 29T^{2} \)
31 \( 1 + 2.02T + 31T^{2} \)
37 \( 1 - 3.52T + 37T^{2} \)
41 \( 1 - 7.60T + 41T^{2} \)
43 \( 1 + 3.42T + 43T^{2} \)
47 \( 1 - 5.77T + 47T^{2} \)
53 \( 1 - 6.56T + 53T^{2} \)
59 \( 1 + 8.44T + 59T^{2} \)
61 \( 1 + 4.08T + 61T^{2} \)
67 \( 1 + 4.70T + 67T^{2} \)
71 \( 1 + 2.73T + 71T^{2} \)
73 \( 1 - 9.55T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + 3.58T + 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + 4.53T + 97T^{2} \)
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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

−7.44709761472335573668923838926, −6.61852313454028115387682014259, −5.77010320349111206848254644698, −5.42049269949853342363487932427, −4.91307765212044886832019107250, −4.20124664403058575871807755505, −2.77885399375041039900290770662, −2.49987608874680220195997829862, −1.43410921545636161905737553360, 0, 1.43410921545636161905737553360, 2.49987608874680220195997829862, 2.77885399375041039900290770662, 4.20124664403058575871807755505, 4.91307765212044886832019107250, 5.42049269949853342363487932427, 5.77010320349111206848254644698, 6.61852313454028115387682014259, 7.44709761472335573668923838926

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