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 − 2.15·3-s + 4-s − 2.78·5-s + 2.15·6-s − 1.33·7-s − 8-s + 1.65·9-s + 2.78·10-s − 5.50·11-s − 2.15·12-s − 5.88·13-s + 1.33·14-s + 5.99·15-s + 16-s − 2.24·17-s − 1.65·18-s − 19-s − 2.78·20-s + 2.87·21-s + 5.50·22-s − 6.52·23-s + 2.15·24-s + 2.72·25-s + 5.88·26-s + 2.90·27-s − 1.33·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.24·3-s + 0.5·4-s − 1.24·5-s + 0.880·6-s − 0.503·7-s − 0.353·8-s + 0.551·9-s + 0.879·10-s − 1.65·11-s − 0.622·12-s − 1.63·13-s + 0.356·14-s + 1.54·15-s + 0.250·16-s − 0.543·17-s − 0.389·18-s − 0.229·19-s − 0.621·20-s + 0.627·21-s + 1.17·22-s − 1.36·23-s + 0.440·24-s + 0.545·25-s + 1.15·26-s + 0.558·27-s − 0.251·28-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 \( 1 + 2.15T + 3T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + 5.50T + 11T^{2} \)
13 \( 1 + 5.88T + 13T^{2} \)
17 \( 1 + 2.24T + 17T^{2} \)
23 \( 1 + 6.52T + 23T^{2} \)
29 \( 1 - 7.17T + 29T^{2} \)
31 \( 1 - 3.81T + 31T^{2} \)
37 \( 1 - 0.00121T + 37T^{2} \)
41 \( 1 + 0.0230T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 1.06T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 - 13.1T + 59T^{2} \)
61 \( 1 - 2.45T + 61T^{2} \)
67 \( 1 - 6.44T + 67T^{2} \)
71 \( 1 - 2.95T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 - 4.54T + 79T^{2} \)
83 \( 1 + 2.00T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 3.39T + 97T^{2} \)
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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

−7.50409371430287434351838281778, −6.84782256869928230503668434026, −6.30043280869611950417423001474, −5.29263059472660488876956031810, −4.89366418382416646258496518672, −4.02914725883962212928013635556, −2.90116352916294722876575106357, −2.26485844172949572618085364352, −0.53864246789753665518813964695, 0, 0.53864246789753665518813964695, 2.26485844172949572618085364352, 2.90116352916294722876575106357, 4.02914725883962212928013635556, 4.89366418382416646258496518672, 5.29263059472660488876956031810, 6.30043280869611950417423001474, 6.84782256869928230503668434026, 7.50409371430287434351838281778

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