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 + 2.76·3-s + 4-s − 0.510·5-s + 2.76·6-s + 4.35·7-s + 8-s + 4.66·9-s − 0.510·10-s + 5.29·11-s + 2.76·12-s + 3.55·13-s + 4.35·14-s − 1.41·15-s + 16-s − 7.99·17-s + 4.66·18-s − 19-s − 0.510·20-s + 12.0·21-s + 5.29·22-s + 1.15·23-s + 2.76·24-s − 4.73·25-s + 3.55·26-s + 4.62·27-s + 4.35·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.59·3-s + 0.5·4-s − 0.228·5-s + 1.13·6-s + 1.64·7-s + 0.353·8-s + 1.55·9-s − 0.161·10-s + 1.59·11-s + 0.799·12-s + 0.985·13-s + 1.16·14-s − 0.365·15-s + 0.250·16-s − 1.93·17-s + 1.10·18-s − 0.229·19-s − 0.114·20-s + 2.62·21-s + 1.12·22-s + 0.240·23-s + 0.565·24-s − 0.947·25-s + 0.696·26-s + 0.889·27-s + 0.822·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $7.962175010$
$L(\frac12)$  $\approx$  $7.962175010$
$L(\frac{3}{2})$   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 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 + 0.510T + 5T^{2} \)
7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 - 5.29T + 11T^{2} \)
13 \( 1 - 3.55T + 13T^{2} \)
17 \( 1 + 7.99T + 17T^{2} \)
23 \( 1 - 1.15T + 23T^{2} \)
29 \( 1 + 1.94T + 29T^{2} \)
31 \( 1 + 8.99T + 31T^{2} \)
37 \( 1 - 8.54T + 37T^{2} \)
41 \( 1 - 3.50T + 41T^{2} \)
43 \( 1 - 9.98T + 43T^{2} \)
47 \( 1 + 5.88T + 47T^{2} \)
53 \( 1 - 10.9T + 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 + 8.35T + 61T^{2} \)
67 \( 1 + 3.77T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 7.77T + 73T^{2} \)
79 \( 1 + 1.43T + 79T^{2} \)
83 \( 1 + 3.11T + 83T^{2} \)
89 \( 1 - 0.187T + 89T^{2} \)
97 \( 1 - 13.0T + 97T^{2} \)
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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

−7.77527066017771033366909007431, −7.37149032122137220837971526619, −6.48471842839290923467630149789, −5.77420134812041039053890925195, −4.59075572655538049367289046645, −4.08306932143139241836312024250, −3.82828688461372327164262998655, −2.65165107033293633692101837211, −1.88894802987437777537699298985, −1.39355113516509952377521246430, 1.39355113516509952377521246430, 1.88894802987437777537699298985, 2.65165107033293633692101837211, 3.82828688461372327164262998655, 4.08306932143139241836312024250, 4.59075572655538049367289046645, 5.77420134812041039053890925195, 6.48471842839290923467630149789, 7.37149032122137220837971526619, 7.77527066017771033366909007431

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