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 + 1.54·3-s + 4-s − 0.0183·5-s + 1.54·6-s − 4.28·7-s + 8-s − 0.618·9-s − 0.0183·10-s + 2.99·11-s + 1.54·12-s + 4.32·13-s − 4.28·14-s − 0.0283·15-s + 16-s − 4.98·17-s − 0.618·18-s + 19-s − 0.0183·20-s − 6.60·21-s + 2.99·22-s − 3.35·23-s + 1.54·24-s − 4.99·25-s + 4.32·26-s − 5.58·27-s − 4.28·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.891·3-s + 0.5·4-s − 0.00821·5-s + 0.630·6-s − 1.61·7-s + 0.353·8-s − 0.206·9-s − 0.00580·10-s + 0.903·11-s + 0.445·12-s + 1.19·13-s − 1.14·14-s − 0.00731·15-s + 0.250·16-s − 1.20·17-s − 0.145·18-s + 0.229·19-s − 0.00410·20-s − 1.44·21-s + 0.638·22-s − 0.699·23-s + 0.315·24-s − 0.999·25-s + 0.847·26-s − 1.07·27-s − 0.809·28-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 \( 1 - 1.54T + 3T^{2} \)
5 \( 1 + 0.0183T + 5T^{2} \)
7 \( 1 + 4.28T + 7T^{2} \)
11 \( 1 - 2.99T + 11T^{2} \)
13 \( 1 - 4.32T + 13T^{2} \)
17 \( 1 + 4.98T + 17T^{2} \)
23 \( 1 + 3.35T + 23T^{2} \)
29 \( 1 - 3.45T + 29T^{2} \)
31 \( 1 + 7.49T + 31T^{2} \)
37 \( 1 - 0.329T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 9.79T + 43T^{2} \)
47 \( 1 + 1.16T + 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 - 1.61T + 61T^{2} \)
67 \( 1 - 10.1T + 67T^{2} \)
71 \( 1 - 4.96T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 8.25T + 79T^{2} \)
83 \( 1 + 3.64T + 83T^{2} \)
89 \( 1 + 8.09T + 89T^{2} \)
97 \( 1 - 2.65T + 97T^{2} \)
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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

−7.41922618223512658667831794473, −6.44589212361473944832852280220, −6.32915134714793805766241421749, −5.57135162909487235332967977964, −4.31205443192592853432348805770, −3.72936785450432740178546349534, −3.30163120309848208653882925361, −2.49628474009486781475183040851, −1.59257913817707638900088634492, 0, 1.59257913817707638900088634492, 2.49628474009486781475183040851, 3.30163120309848208653882925361, 3.72936785450432740178546349534, 4.31205443192592853432348805770, 5.57135162909487235332967977964, 6.32915134714793805766241421749, 6.44589212361473944832852280220, 7.41922618223512658667831794473

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