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 + 1.62·3-s + 4-s − 1.52·5-s + 1.62·6-s − 1.42·7-s + 8-s − 0.344·9-s − 1.52·10-s − 3.50·11-s + 1.62·12-s − 0.327·13-s − 1.42·14-s − 2.47·15-s + 16-s + 7.06·17-s − 0.344·18-s + 19-s − 1.52·20-s − 2.32·21-s − 3.50·22-s + 3.94·23-s + 1.62·24-s − 2.68·25-s − 0.327·26-s − 5.45·27-s − 1.42·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.940·3-s + 0.5·4-s − 0.679·5-s + 0.665·6-s − 0.538·7-s + 0.353·8-s − 0.114·9-s − 0.480·10-s − 1.05·11-s + 0.470·12-s − 0.0907·13-s − 0.381·14-s − 0.639·15-s + 0.250·16-s + 1.71·17-s − 0.0812·18-s + 0.229·19-s − 0.339·20-s − 0.506·21-s − 0.746·22-s + 0.822·23-s + 0.332·24-s − 0.537·25-s − 0.0641·26-s − 1.04·27-s − 0.269·28-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 \( 1 - 1.62T + 3T^{2} \)
5 \( 1 + 1.52T + 5T^{2} \)
7 \( 1 + 1.42T + 7T^{2} \)
11 \( 1 + 3.50T + 11T^{2} \)
13 \( 1 + 0.327T + 13T^{2} \)
17 \( 1 - 7.06T + 17T^{2} \)
23 \( 1 - 3.94T + 23T^{2} \)
29 \( 1 + 4.95T + 29T^{2} \)
31 \( 1 - 4.87T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 - 4.65T + 41T^{2} \)
43 \( 1 - 5.36T + 43T^{2} \)
47 \( 1 + 6.86T + 47T^{2} \)
53 \( 1 + 7.45T + 53T^{2} \)
59 \( 1 + 7.68T + 59T^{2} \)
61 \( 1 - 11.7T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 1.58T + 71T^{2} \)
73 \( 1 + 6.38T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 11.7T + 89T^{2} \)
97 \( 1 + 6.31T + 97T^{2} \)
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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

−7.59112457087076986855210917251, −6.97913436680789017311342874912, −5.85289358358215912328450066458, −5.47203934280228171147505011953, −4.54726662622284380581119269131, −3.68170754600413725129144181901, −3.09480827605337953034909348922, −2.72571419847979009497537563750, −1.50487334306441104995165970202, 0, 1.50487334306441104995165970202, 2.72571419847979009497537563750, 3.09480827605337953034909348922, 3.68170754600413725129144181901, 4.54726662622284380581119269131, 5.47203934280228171147505011953, 5.85289358358215912328450066458, 6.97913436680789017311342874912, 7.59112457087076986855210917251

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