Properties

Degree 2
Conductor $ 2 \cdot 7^{2} \cdot 41 $
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 − 3-s + 4-s + 3·5-s + 6-s − 8-s − 2·9-s − 3·10-s − 12-s − 2·13-s − 3·15-s + 16-s + 3·17-s + 2·18-s + 4·19-s + 3·20-s − 6·23-s + 24-s + 4·25-s + 2·26-s + 5·27-s − 3·29-s + 3·30-s + 31-s − 32-s − 3·34-s − 2·36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.948·10-s − 0.288·12-s − 0.554·13-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s + 0.917·19-s + 0.670·20-s − 1.25·23-s + 0.204·24-s + 4/5·25-s + 0.392·26-s + 0.962·27-s − 0.557·29-s + 0.547·30-s + 0.179·31-s − 0.176·32-s − 0.514·34-s − 1/3·36-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  =  0
Selberg data  =  $(2,\ 4018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.233855249$
$L(\frac12)$  $\approx$  $1.233855249$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
7 \( 1 \)
41 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−8.520556370020037803487923175027, −7.76037107809754618468067160120, −6.93521902518121016458251189993, −6.17434638576743097650084240260, −5.54491032545906079293187173263, −5.15921754473917111610804092803, −3.69695273217897586311341032517, −2.62416708962628356954759718551, −1.89316458112345223945003256981, −0.72259802345287277157643649594, 0.72259802345287277157643649594, 1.89316458112345223945003256981, 2.62416708962628356954759718551, 3.69695273217897586311341032517, 5.15921754473917111610804092803, 5.54491032545906079293187173263, 6.17434638576743097650084240260, 6.93521902518121016458251189993, 7.76037107809754618468067160120, 8.520556370020037803487923175027

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