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 + 1.66·3-s + 4-s − 2.16·5-s + 1.66·6-s + 3.52·7-s + 8-s − 0.222·9-s − 2.16·10-s − 3.93·11-s + 1.66·12-s − 6.14·13-s + 3.52·14-s − 3.60·15-s + 16-s + 5.51·17-s − 0.222·18-s + 19-s − 2.16·20-s + 5.87·21-s − 3.93·22-s − 0.382·23-s + 1.66·24-s − 0.310·25-s − 6.14·26-s − 5.37·27-s + 3.52·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.962·3-s + 0.5·4-s − 0.968·5-s + 0.680·6-s + 1.33·7-s + 0.353·8-s − 0.0741·9-s − 0.684·10-s − 1.18·11-s + 0.481·12-s − 1.70·13-s + 0.942·14-s − 0.931·15-s + 0.250·16-s + 1.33·17-s − 0.0524·18-s + 0.229·19-s − 0.484·20-s + 1.28·21-s − 0.839·22-s − 0.0797·23-s + 0.340·24-s − 0.0621·25-s − 1.20·26-s − 1.03·27-s + 0.666·28-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 \( 1 - 1.66T + 3T^{2} \)
5 \( 1 + 2.16T + 5T^{2} \)
7 \( 1 - 3.52T + 7T^{2} \)
11 \( 1 + 3.93T + 11T^{2} \)
13 \( 1 + 6.14T + 13T^{2} \)
17 \( 1 - 5.51T + 17T^{2} \)
23 \( 1 + 0.382T + 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 + 5.31T + 31T^{2} \)
37 \( 1 - 4.36T + 37T^{2} \)
41 \( 1 + 7.85T + 41T^{2} \)
43 \( 1 + 3.27T + 43T^{2} \)
47 \( 1 + 3.34T + 47T^{2} \)
53 \( 1 + 5.13T + 53T^{2} \)
59 \( 1 - 0.712T + 59T^{2} \)
61 \( 1 + 3.63T + 61T^{2} \)
67 \( 1 - 1.30T + 67T^{2} \)
71 \( 1 + 2.12T + 71T^{2} \)
73 \( 1 + 13.5T + 73T^{2} \)
79 \( 1 + 2.21T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 1.18T + 89T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−7.61176899266132268070903623243, −7.22407691865209094831430887818, −5.85165050449518062401826559786, −5.05529176863675809091399820066, −4.82351684869237956698001928995, −3.81364027766625603707903269600, −3.11575388998611551351779783055, −2.47778391316029153745985954122, −1.64077988517538465786514690808, 0, 1.64077988517538465786514690808, 2.47778391316029153745985954122, 3.11575388998611551351779783055, 3.81364027766625603707903269600, 4.82351684869237956698001928995, 5.05529176863675809091399820066, 5.85165050449518062401826559786, 7.22407691865209094831430887818, 7.61176899266132268070903623243

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