Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.256·3-s + 4-s + 4.11·5-s − 0.256·6-s + 0.895·7-s + 8-s − 2.93·9-s + 4.11·10-s + 3.71·11-s − 0.256·12-s + 5.72·13-s + 0.895·14-s − 1.05·15-s + 16-s + 7.80·17-s − 2.93·18-s − 19-s + 4.11·20-s − 0.230·21-s + 3.71·22-s − 5.69·23-s − 0.256·24-s + 11.9·25-s + 5.72·26-s + 1.52·27-s + 0.895·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.148·3-s + 0.5·4-s + 1.83·5-s − 0.104·6-s + 0.338·7-s + 0.353·8-s − 0.978·9-s + 1.30·10-s + 1.12·11-s − 0.0741·12-s + 1.58·13-s + 0.239·14-s − 0.272·15-s + 0.250·16-s + 1.89·17-s − 0.691·18-s − 0.229·19-s + 0.919·20-s − 0.0502·21-s + 0.792·22-s − 1.18·23-s − 0.0524·24-s + 2.38·25-s + 1.12·26-s + 0.293·27-s + 0.169·28-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  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.700579504$
$L(\frac12)$  $\approx$  $5.700579504$
$L(\frac{3}{2})$   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 \( 1 + 0.256T + 3T^{2} \)
5 \( 1 - 4.11T + 5T^{2} \)
7 \( 1 - 0.895T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 - 5.72T + 13T^{2} \)
17 \( 1 - 7.80T + 17T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 - 1.52T + 29T^{2} \)
31 \( 1 - 5.49T + 31T^{2} \)
37 \( 1 - 9.54T + 37T^{2} \)
41 \( 1 + 2.67T + 41T^{2} \)
43 \( 1 + 5.87T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 9.27T + 53T^{2} \)
59 \( 1 + 13.9T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 6.70T + 67T^{2} \)
71 \( 1 + 14.1T + 71T^{2} \)
73 \( 1 + 8.71T + 73T^{2} \)
79 \( 1 + 9.54T + 79T^{2} \)
83 \( 1 + 8.82T + 83T^{2} \)
89 \( 1 - 6.97T + 89T^{2} \)
97 \( 1 - 9.18T + 97T^{2} \)
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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

−8.014380610712741514336060641806, −6.60115060929940039570213414314, −6.22914962120939169877234845602, −5.88182695185003447529224521274, −5.24593696740991706406030244523, −4.39263538927710861797704334806, −3.36935960192808066477930314412, −2.83397584239500825758692786044, −1.59756888818674776871257415257, −1.30317271462565662851335735647, 1.30317271462565662851335735647, 1.59756888818674776871257415257, 2.83397584239500825758692786044, 3.36935960192808066477930314412, 4.39263538927710861797704334806, 5.24593696740991706406030244523, 5.88182695185003447529224521274, 6.22914962120939169877234845602, 6.60115060929940039570213414314, 8.014380610712741514336060641806

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