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 + 0.100·3-s + 4-s + 2.02·5-s − 0.100·6-s + 1.91·7-s − 8-s − 2.98·9-s − 2.02·10-s − 3.03·11-s + 0.100·12-s − 1.59·13-s − 1.91·14-s + 0.203·15-s + 16-s + 4.83·17-s + 2.98·18-s − 19-s + 2.02·20-s + 0.192·21-s + 3.03·22-s + 2.15·23-s − 0.100·24-s − 0.893·25-s + 1.59·26-s − 0.602·27-s + 1.91·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0580·3-s + 0.5·4-s + 0.906·5-s − 0.0410·6-s + 0.722·7-s − 0.353·8-s − 0.996·9-s − 0.640·10-s − 0.915·11-s + 0.0290·12-s − 0.441·13-s − 0.511·14-s + 0.0526·15-s + 0.250·16-s + 1.17·17-s + 0.704·18-s − 0.229·19-s + 0.453·20-s + 0.0419·21-s + 0.647·22-s + 0.449·23-s − 0.0205·24-s − 0.178·25-s + 0.312·26-s − 0.115·27-s + 0.361·28-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 \( 1 - 0.100T + 3T^{2} \)
5 \( 1 - 2.02T + 5T^{2} \)
7 \( 1 - 1.91T + 7T^{2} \)
11 \( 1 + 3.03T + 11T^{2} \)
13 \( 1 + 1.59T + 13T^{2} \)
17 \( 1 - 4.83T + 17T^{2} \)
23 \( 1 - 2.15T + 23T^{2} \)
29 \( 1 + 1.54T + 29T^{2} \)
31 \( 1 + 3.26T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 0.0525T + 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 9.77T + 53T^{2} \)
59 \( 1 - 3.74T + 59T^{2} \)
61 \( 1 - 9.52T + 61T^{2} \)
67 \( 1 - 11.6T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 5.61T + 73T^{2} \)
79 \( 1 + 8.77T + 79T^{2} \)
83 \( 1 + 4.40T + 83T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 - 17.8T + 97T^{2} \)
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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

−7.60071200170569905195994673128, −7.00571295365032149015416010649, −5.93708182310119089115001501498, −5.50304012587479175446962505602, −5.02061219287184057936425824871, −3.72502841329004128209302931562, −2.76761830767617465781954387983, −2.19972126684921517548816560619, −1.30103986097828195048975062297, 0, 1.30103986097828195048975062297, 2.19972126684921517548816560619, 2.76761830767617465781954387983, 3.72502841329004128209302931562, 5.02061219287184057936425824871, 5.50304012587479175446962505602, 5.93708182310119089115001501498, 7.00571295365032149015416010649, 7.60071200170569905195994673128

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