Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 1.51·3-s + 4-s − 1.33·5-s − 1.51·6-s − 3.41·7-s + 8-s − 0.705·9-s − 1.33·10-s − 1.69·11-s − 1.51·12-s − 1.37·13-s − 3.41·14-s + 2.01·15-s + 16-s + 3.29·17-s − 0.705·18-s − 19-s − 1.33·20-s + 5.16·21-s − 1.69·22-s − 9.07·23-s − 1.51·24-s − 3.22·25-s − 1.37·26-s + 5.61·27-s − 3.41·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.874·3-s + 0.5·4-s − 0.594·5-s − 0.618·6-s − 1.28·7-s + 0.353·8-s − 0.235·9-s − 0.420·10-s − 0.512·11-s − 0.437·12-s − 0.381·13-s − 0.912·14-s + 0.520·15-s + 0.250·16-s + 0.799·17-s − 0.166·18-s − 0.229·19-s − 0.297·20-s + 1.12·21-s − 0.362·22-s − 1.89·23-s − 0.309·24-s − 0.645·25-s − 0.269·26-s + 1.08·27-s − 0.644·28-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  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4179489258$
$L(\frac12)$  $\approx$  $0.4179489258$
$L(\frac{3}{2})$   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 \( 1 + 1.51T + 3T^{2} \)
5 \( 1 + 1.33T + 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 1.69T + 11T^{2} \)
13 \( 1 + 1.37T + 13T^{2} \)
17 \( 1 - 3.29T + 17T^{2} \)
23 \( 1 + 9.07T + 23T^{2} \)
29 \( 1 + 4.00T + 29T^{2} \)
31 \( 1 + 3.30T + 31T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
41 \( 1 - 3.82T + 41T^{2} \)
43 \( 1 + 2.87T + 43T^{2} \)
47 \( 1 - 0.403T + 47T^{2} \)
53 \( 1 + 3.47T + 53T^{2} \)
59 \( 1 + 4.97T + 59T^{2} \)
61 \( 1 - 5.81T + 61T^{2} \)
67 \( 1 - 2.62T + 67T^{2} \)
71 \( 1 + 1.72T + 71T^{2} \)
73 \( 1 + 2.94T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 0.930T + 89T^{2} \)
97 \( 1 - 19.0T + 97T^{2} \)
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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

−7.62772273274959311684048477253, −7.01934461528663616455434517000, −6.17243312278070017756997352913, −5.81251456232009463192142848344, −5.20004441363032903817556136081, −4.24260411907468741992914528078, −3.58655297990964335806155694932, −2.93599555216577329901167928122, −1.90226712690623792675605852810, −0.28006036995578447895045119442, 0.28006036995578447895045119442, 1.90226712690623792675605852810, 2.93599555216577329901167928122, 3.58655297990964335806155694932, 4.24260411907468741992914528078, 5.20004441363032903817556136081, 5.81251456232009463192142848344, 6.17243312278070017756997352913, 7.01934461528663616455434517000, 7.62772273274959311684048477253

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