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 − 1.71·3-s + 4-s + 1.96·5-s + 1.71·6-s + 0.664·7-s − 8-s − 0.0513·9-s − 1.96·10-s + 5.44·11-s − 1.71·12-s − 0.0515·13-s − 0.664·14-s − 3.37·15-s + 16-s − 7.45·17-s + 0.0513·18-s − 19-s + 1.96·20-s − 1.14·21-s − 5.44·22-s − 0.569·23-s + 1.71·24-s − 1.13·25-s + 0.0515·26-s + 5.23·27-s + 0.664·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.991·3-s + 0.5·4-s + 0.879·5-s + 0.701·6-s + 0.251·7-s − 0.353·8-s − 0.0171·9-s − 0.621·10-s + 1.64·11-s − 0.495·12-s − 0.0143·13-s − 0.177·14-s − 0.871·15-s + 0.250·16-s − 1.80·17-s + 0.0121·18-s − 0.229·19-s + 0.439·20-s − 0.249·21-s − 1.16·22-s − 0.118·23-s + 0.350·24-s − 0.226·25-s + 0.0101·26-s + 1.00·27-s + 0.125·28-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 \( 1 + 1.71T + 3T^{2} \)
5 \( 1 - 1.96T + 5T^{2} \)
7 \( 1 - 0.664T + 7T^{2} \)
11 \( 1 - 5.44T + 11T^{2} \)
13 \( 1 + 0.0515T + 13T^{2} \)
17 \( 1 + 7.45T + 17T^{2} \)
23 \( 1 + 0.569T + 23T^{2} \)
29 \( 1 + 2.76T + 29T^{2} \)
31 \( 1 - 4.18T + 31T^{2} \)
37 \( 1 - 3.80T + 37T^{2} \)
41 \( 1 - 3.76T + 41T^{2} \)
43 \( 1 - 5.34T + 43T^{2} \)
47 \( 1 - 2.90T + 47T^{2} \)
53 \( 1 + 4.82T + 53T^{2} \)
59 \( 1 + 0.401T + 59T^{2} \)
61 \( 1 + 7.28T + 61T^{2} \)
67 \( 1 + 11.1T + 67T^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + 0.623T + 73T^{2} \)
79 \( 1 + 3.96T + 79T^{2} \)
83 \( 1 - 7.33T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 1.39T + 97T^{2} \)
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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

−7.36513223907952866052888468715, −6.55022137264758150917006636797, −6.23934672790005764495654920661, −5.74444774474351837808905416934, −4.66979482961666937725899958373, −4.13828800656914265376131885833, −2.82883798211946360620274932507, −1.93395474034878935681888361170, −1.19260927561614373807516026961, 0, 1.19260927561614373807516026961, 1.93395474034878935681888361170, 2.82883798211946360620274932507, 4.13828800656914265376131885833, 4.66979482961666937725899958373, 5.74444774474351837808905416934, 6.23934672790005764495654920661, 6.55022137264758150917006636797, 7.36513223907952866052888468715

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