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 − 2.46·3-s + 4-s − 0.0638·5-s + 2.46·6-s + 3.24·7-s − 8-s + 3.06·9-s + 0.0638·10-s + 1.81·11-s − 2.46·12-s − 6.18·13-s − 3.24·14-s + 0.157·15-s + 16-s + 2.99·17-s − 3.06·18-s − 19-s − 0.0638·20-s − 7.98·21-s − 1.81·22-s − 0.142·23-s + 2.46·24-s − 4.99·25-s + 6.18·26-s − 0.171·27-s + 3.24·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.0285·5-s + 1.00·6-s + 1.22·7-s − 0.353·8-s + 1.02·9-s + 0.0201·10-s + 0.547·11-s − 0.711·12-s − 1.71·13-s − 0.866·14-s + 0.0405·15-s + 0.250·16-s + 0.726·17-s − 0.723·18-s − 0.229·19-s − 0.0142·20-s − 1.74·21-s − 0.386·22-s − 0.0296·23-s + 0.502·24-s − 0.999·25-s + 1.21·26-s − 0.0329·27-s + 0.612·28-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 \( 1 + 2.46T + 3T^{2} \)
5 \( 1 + 0.0638T + 5T^{2} \)
7 \( 1 - 3.24T + 7T^{2} \)
11 \( 1 - 1.81T + 11T^{2} \)
13 \( 1 + 6.18T + 13T^{2} \)
17 \( 1 - 2.99T + 17T^{2} \)
23 \( 1 + 0.142T + 23T^{2} \)
29 \( 1 - 1.88T + 29T^{2} \)
31 \( 1 - 8.38T + 31T^{2} \)
37 \( 1 + 0.447T + 37T^{2} \)
41 \( 1 + 10.3T + 41T^{2} \)
43 \( 1 - 9.43T + 43T^{2} \)
47 \( 1 + 4.96T + 47T^{2} \)
53 \( 1 - 3.48T + 53T^{2} \)
59 \( 1 - 2.98T + 59T^{2} \)
61 \( 1 - 4.57T + 61T^{2} \)
67 \( 1 + 6.70T + 67T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 3.92T + 79T^{2} \)
83 \( 1 - 4.00T + 83T^{2} \)
89 \( 1 + 8.86T + 89T^{2} \)
97 \( 1 - 2.20T + 97T^{2} \)
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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

−7.45457405495702055928523388069, −6.88057162489741451494404331545, −6.09557138375964308131261467136, −5.44945219836996657155304837222, −4.80080454824369754978436680033, −4.25690955654103478347112493599, −2.87822884703383775170015149825, −1.87930957825620555274784426081, −1.05335987817910327696220501808, 0, 1.05335987817910327696220501808, 1.87930957825620555274784426081, 2.87822884703383775170015149825, 4.25690955654103478347112493599, 4.80080454824369754978436680033, 5.44945219836996657155304837222, 6.09557138375964308131261467136, 6.88057162489741451494404331545, 7.45457405495702055928523388069

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