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 + 1.77·3-s + 4-s + 0.642·5-s + 1.77·6-s + 4.80·7-s + 8-s + 0.150·9-s + 0.642·10-s + 3.51·11-s + 1.77·12-s − 0.225·13-s + 4.80·14-s + 1.14·15-s + 16-s + 0.444·17-s + 0.150·18-s − 19-s + 0.642·20-s + 8.52·21-s + 3.51·22-s − 2.47·23-s + 1.77·24-s − 4.58·25-s − 0.225·26-s − 5.05·27-s + 4.80·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.02·3-s + 0.5·4-s + 0.287·5-s + 0.724·6-s + 1.81·7-s + 0.353·8-s + 0.0501·9-s + 0.203·10-s + 1.05·11-s + 0.512·12-s − 0.0624·13-s + 1.28·14-s + 0.294·15-s + 0.250·16-s + 0.107·17-s + 0.0354·18-s − 0.229·19-s + 0.143·20-s + 1.86·21-s + 0.748·22-s − 0.515·23-s + 0.362·24-s − 0.917·25-s − 0.0441·26-s − 0.973·27-s + 0.908·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.826708280$
$L(\frac12)$  $\approx$  $6.826708280$
$L(\frac{3}{2})$   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 \( 1 - 1.77T + 3T^{2} \)
5 \( 1 - 0.642T + 5T^{2} \)
7 \( 1 - 4.80T + 7T^{2} \)
11 \( 1 - 3.51T + 11T^{2} \)
13 \( 1 + 0.225T + 13T^{2} \)
17 \( 1 - 0.444T + 17T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 - 0.426T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 - 2.73T + 37T^{2} \)
41 \( 1 + 6.26T + 41T^{2} \)
43 \( 1 + 6.25T + 43T^{2} \)
47 \( 1 - 2.41T + 47T^{2} \)
53 \( 1 - 0.377T + 53T^{2} \)
59 \( 1 - 14.3T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 11.9T + 67T^{2} \)
71 \( 1 + 9.01T + 71T^{2} \)
73 \( 1 + 8.40T + 73T^{2} \)
79 \( 1 + 4.63T + 79T^{2} \)
83 \( 1 - 5.87T + 83T^{2} \)
89 \( 1 - 1.43T + 89T^{2} \)
97 \( 1 + 3.02T + 97T^{2} \)
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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

−7.907883930007513882105734665407, −7.25692933292939019737911762749, −6.36603455899288883982085940766, −5.67390363635678623964894993760, −4.88783346533649690493500340926, −4.22868517980449106912700972670, −3.64230013976414027432634195884, −2.59201802361864995181463816665, −1.97692176485767944951250947861, −1.24596620058221718437775254843, 1.24596620058221718437775254843, 1.97692176485767944951250947861, 2.59201802361864995181463816665, 3.64230013976414027432634195884, 4.22868517980449106912700972670, 4.88783346533649690493500340926, 5.67390363635678623964894993760, 6.36603455899288883982085940766, 7.25692933292939019737911762749, 7.907883930007513882105734665407

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