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 + 0.994·3-s + 4-s − 3.18·5-s − 0.994·6-s − 0.721·7-s − 8-s − 2.01·9-s + 3.18·10-s − 1.88·11-s + 0.994·12-s + 1.86·13-s + 0.721·14-s − 3.17·15-s + 16-s + 2.19·17-s + 2.01·18-s − 19-s − 3.18·20-s − 0.717·21-s + 1.88·22-s + 3.75·23-s − 0.994·24-s + 5.16·25-s − 1.86·26-s − 4.98·27-s − 0.721·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.574·3-s + 0.5·4-s − 1.42·5-s − 0.406·6-s − 0.272·7-s − 0.353·8-s − 0.670·9-s + 1.00·10-s − 0.568·11-s + 0.287·12-s + 0.516·13-s + 0.192·14-s − 0.818·15-s + 0.250·16-s + 0.532·17-s + 0.473·18-s − 0.229·19-s − 0.712·20-s − 0.156·21-s + 0.402·22-s + 0.782·23-s − 0.203·24-s + 1.03·25-s − 0.365·26-s − 0.959·27-s − 0.136·28-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 \( 1 - 0.994T + 3T^{2} \)
5 \( 1 + 3.18T + 5T^{2} \)
7 \( 1 + 0.721T + 7T^{2} \)
11 \( 1 + 1.88T + 11T^{2} \)
13 \( 1 - 1.86T + 13T^{2} \)
17 \( 1 - 2.19T + 17T^{2} \)
23 \( 1 - 3.75T + 23T^{2} \)
29 \( 1 - 5.55T + 29T^{2} \)
31 \( 1 - 2.69T + 31T^{2} \)
37 \( 1 + 5.33T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 2.68T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 + 0.302T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 - 0.637T + 61T^{2} \)
67 \( 1 + 3.50T + 67T^{2} \)
71 \( 1 - 1.48T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + 5.62T + 83T^{2} \)
89 \( 1 - 5.68T + 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−7.81493947446082763603023445566, −7.00532936527149352423487273985, −6.34840368937934946021842647325, −5.41398364432637087500176941174, −4.56651529253380421994428103262, −3.56594964096731115599252613352, −3.16505171426808491023315401011, −2.35429947166293418669597717488, −1.00133975522981844673510664130, 0, 1.00133975522981844673510664130, 2.35429947166293418669597717488, 3.16505171426808491023315401011, 3.56594964096731115599252613352, 4.56651529253380421994428103262, 5.41398364432637087500176941174, 6.34840368937934946021842647325, 7.00532936527149352423487273985, 7.81493947446082763603023445566

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