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 + 2.08·3-s + 4-s − 0.817·5-s + 2.08·6-s − 0.777·7-s + 8-s + 1.35·9-s − 0.817·10-s + 0.583·11-s + 2.08·12-s + 6.54·13-s − 0.777·14-s − 1.70·15-s + 16-s − 2.73·17-s + 1.35·18-s − 19-s − 0.817·20-s − 1.62·21-s + 0.583·22-s + 6.31·23-s + 2.08·24-s − 4.33·25-s + 6.54·26-s − 3.42·27-s − 0.777·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.20·3-s + 0.5·4-s − 0.365·5-s + 0.852·6-s − 0.293·7-s + 0.353·8-s + 0.453·9-s − 0.258·10-s + 0.175·11-s + 0.602·12-s + 1.81·13-s − 0.207·14-s − 0.440·15-s + 0.250·16-s − 0.664·17-s + 0.320·18-s − 0.229·19-s − 0.182·20-s − 0.354·21-s + 0.124·22-s + 1.31·23-s + 0.426·24-s − 0.866·25-s + 1.28·26-s − 0.659·27-s − 0.146·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.223446095$
$L(\frac12)$  $\approx$  $5.223446095$
$L(\frac{3}{2})$   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 \( 1 - 2.08T + 3T^{2} \)
5 \( 1 + 0.817T + 5T^{2} \)
7 \( 1 + 0.777T + 7T^{2} \)
11 \( 1 - 0.583T + 11T^{2} \)
13 \( 1 - 6.54T + 13T^{2} \)
17 \( 1 + 2.73T + 17T^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 - 5.06T + 29T^{2} \)
31 \( 1 - 8.73T + 31T^{2} \)
37 \( 1 - 0.521T + 37T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 + 7.30T + 47T^{2} \)
53 \( 1 - 2.58T + 53T^{2} \)
59 \( 1 + 6.28T + 59T^{2} \)
61 \( 1 - 3.79T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + 5.81T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 - 4.08T + 89T^{2} \)
97 \( 1 + 11.4T + 97T^{2} \)
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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

−8.005767128418230107118382164530, −7.11354831996893233422912223235, −6.35418253043370523396925347814, −5.94815051027155967533187188430, −4.73784564568850633235801950628, −4.16318074140235745737141681369, −3.39105869280488386644637797937, −2.97633562481856728848608596996, −2.05420241396626604425453505399, −0.994585898524853195793548759160, 0.994585898524853195793548759160, 2.05420241396626604425453505399, 2.97633562481856728848608596996, 3.39105869280488386644637797937, 4.16318074140235745737141681369, 4.73784564568850633235801950628, 5.94815051027155967533187188430, 6.35418253043370523396925347814, 7.11354831996893233422912223235, 8.005767128418230107118382164530

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