Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 41 $
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 + 2·3-s + 4-s − 4·5-s − 2·6-s − 8-s + 9-s + 4·10-s + 4·11-s + 2·12-s − 4·13-s − 8·15-s + 16-s + 2·17-s − 18-s + 6·19-s − 4·20-s − 4·22-s − 8·23-s − 2·24-s + 11·25-s + 4·26-s − 4·27-s + 6·29-s + 8·30-s − 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 1.78·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.20·11-s + 0.577·12-s − 1.10·13-s − 2.06·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.894·20-s − 0.852·22-s − 1.66·23-s − 0.408·24-s + 11/5·25-s + 0.784·26-s − 0.769·27-s + 1.11·29-s + 1.46·30-s − 0.718·31-s − 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4018\)    =    \(2 \cdot 7^{2} \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 4018,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−8.106419331134717351209648986269, −7.54341270905930009265662143965, −7.15883087682184774862670323176, −6.09442306771597635870652030308, −4.82297994473093773677311759830, −3.92776200269073340938046439556, −3.38478078744207192029083494001, −2.60641153083624763490742346794, −1.33128905378050739229027921683, 0, 1.33128905378050739229027921683, 2.60641153083624763490742346794, 3.38478078744207192029083494001, 3.92776200269073340938046439556, 4.82297994473093773677311759830, 6.09442306771597635870652030308, 7.15883087682184774862670323176, 7.54341270905930009265662143965, 8.106419331134717351209648986269

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