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 + 3.03·3-s + 4-s − 2.17·5-s + 3.03·6-s + 4.19·7-s + 8-s + 6.22·9-s − 2.17·10-s − 5.40·11-s + 3.03·12-s + 3.05·13-s + 4.19·14-s − 6.60·15-s + 16-s + 1.20·17-s + 6.22·18-s − 19-s − 2.17·20-s + 12.7·21-s − 5.40·22-s + 2.56·23-s + 3.03·24-s − 0.263·25-s + 3.05·26-s + 9.78·27-s + 4.19·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.75·3-s + 0.5·4-s − 0.973·5-s + 1.23·6-s + 1.58·7-s + 0.353·8-s + 2.07·9-s − 0.688·10-s − 1.63·11-s + 0.876·12-s + 0.847·13-s + 1.12·14-s − 1.70·15-s + 0.250·16-s + 0.291·17-s + 1.46·18-s − 0.229·19-s − 0.486·20-s + 2.78·21-s − 1.15·22-s + 0.535·23-s + 0.619·24-s − 0.0527·25-s + 0.599·26-s + 1.88·27-s + 0.793·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.722934302$
$L(\frac12)$  $\approx$  $6.722934302$
$L(\frac{3}{2})$   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 \( 1 - 3.03T + 3T^{2} \)
5 \( 1 + 2.17T + 5T^{2} \)
7 \( 1 - 4.19T + 7T^{2} \)
11 \( 1 + 5.40T + 11T^{2} \)
13 \( 1 - 3.05T + 13T^{2} \)
17 \( 1 - 1.20T + 17T^{2} \)
23 \( 1 - 2.56T + 23T^{2} \)
29 \( 1 - 7.69T + 29T^{2} \)
31 \( 1 - 7.26T + 31T^{2} \)
37 \( 1 + 4.39T + 37T^{2} \)
41 \( 1 + 9.33T + 41T^{2} \)
43 \( 1 - 0.661T + 43T^{2} \)
47 \( 1 - 0.0963T + 47T^{2} \)
53 \( 1 - 7.95T + 53T^{2} \)
59 \( 1 - 1.30T + 59T^{2} \)
61 \( 1 + 0.554T + 61T^{2} \)
67 \( 1 - 5.21T + 67T^{2} \)
71 \( 1 - 0.750T + 71T^{2} \)
73 \( 1 - 7.23T + 73T^{2} \)
79 \( 1 - 7.40T + 79T^{2} \)
83 \( 1 + 0.203T + 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 0.998T + 97T^{2} \)
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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

−8.029688766751581782688885538490, −7.40946768069460220410356866913, −6.71281524648170588629295035659, −5.41126876575012772969291209756, −4.80002696153107150331424690434, −4.21183450964889869514491042256, −3.47781132080792545756576682557, −2.79026001530098968643426107771, −2.11953703676064787214836404738, −1.14237220854323622818004085693, 1.14237220854323622818004085693, 2.11953703676064787214836404738, 2.79026001530098968643426107771, 3.47781132080792545756576682557, 4.21183450964889869514491042256, 4.80002696153107150331424690434, 5.41126876575012772969291209756, 6.71281524648170588629295035659, 7.40946768069460220410356866913, 8.029688766751581782688885538490

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